This page collects the integrals class 12 formulas from the NCERT Maths Chapter 7, Integrals — the standard integral table, the properties of indefinite integrals, substitution, partial fractions, integration by parts, and definite-integral properties. It is built for quick revision: find the formula, check its condition, and note what each symbol means.
Every formula is grouped by its textbook topic, with the meaning of each symbol, when to use it, and worked examples using fresh numbers. For the full explanations and derivations, visit the Class 12 Maths formulas hub. You can verify any result directly in the official NCERT Integrals chapter PDF (lemh201.pdf).
Formulas at a Glance
The index below covers the formulas grouped later on this page — use it to find a result fast, then read its conditions in the grouped list.
| Purpose (what you are finding) | Formula |
|---|---|
| Power rule (integral of any power of x) | \( \int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\ n \neq -1;\quad \int dx = x + C \) |
| Reciprocal of x — the n = −1 case | \( \int \frac{1}{x}\,dx = \log|x| + C \) |
| Exponential functions | \( \int e^x\,dx = e^x + C;\quad \int a^x\,dx = \frac{a^x}{\log a} + C \) |
| Sine and cosine | \( \int \sin x\,dx = -\cos x + C;\quad \int \cos x\,dx = \sin x + C \) |
| Derivative-based trig forms (sec², cosec², sec·tan, cosec·cot) | \( \int \sec^2 x\,dx = \tan x + C;\ \int \operatorname{cosec}^2 x\,dx = -\cot x + C;\ \int \sec x\tan x\,dx = \sec x + C;\ \int \operatorname{cosec} x\cot x\,dx = -\operatorname{cosec} x + C \) |
| Inverse-trig results | \( \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1} x + C;\quad \int \frac{dx}{1+x^2} = \tan^{-1} x + C \) |
| tan x, cot x, sec x, cosec x (log forms) | \( \int \tan x\,dx = \log|\sec x| + C;\ \int \cot x\,dx = \log|\sin x| + C;\ \int \sec x\,dx = \log|\sec x + \tan x| + C;\ \int \operatorname{cosec} x\,dx = \log|\operatorname{cosec} x – \cot x| + C \) |
| Reciprocal of a difference of squares | \( \int \frac{dx}{x^2-a^2} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right| + C;\quad \int \frac{dx}{a^2-x^2} = \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right| + C \) |
| Reciprocal of a sum of squares | \( \int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C \) |
| Reciprocal of square roots of quadratics | \( \int \frac{dx}{\sqrt{x^2-a^2}} = \log\left|x+\sqrt{x^2-a^2}\right| + C;\ \int \frac{dx}{\sqrt{x^2+a^2}} = \log\left|x+\sqrt{x^2+a^2}\right| + C;\ \int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a} + C \) |
| Substitution (change of variable) | \( x = g(t),\ dx = g'(t)\,dt \Rightarrow \int f(x)\,dx = \int f(g(t))\,g'(t)\,dt \) |
| Integration by parts (product of two functions) | \( \int f(x)g(x)\,dx = f(x)\int g(x)\,dx – \int\left[f'(x)\int g(x)\,dx\right]dx \) |
| Special exponential form — eˣ times (f + f′) | \( \int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C \) |
| Square-root integrals (from by parts) | \( \int \sqrt{x^2-a^2}\,dx = \frac{x}{2}\sqrt{x^2-a^2} – \frac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right| + C;\ \int \sqrt{x^2+a^2}\,dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log\left|x+\sqrt{x^2+a^2}\right| + C;\ \int \sqrt{a^2-x^2}\,dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C \) |
| Definite integral value (second fundamental theorem) | \( \int_a^b f(x)\,dx = [F(x)]_a^b = F(b) – F(a),\quad F'(x) = f(x) \) |
| Symmetry properties for limits 0→2a and −a→a | \( \int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx \) if \( f(2a-x) = f(x) \), and is 0 if \( f(2a-x) = -f(x) \); \( \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \) for f even, 0 for f odd. |
All Formulas, Grouped by Topic
The formulas below are grouped under the sub-topics where the NCERT textbook presents them. Page references are to the NCERT Class 12 Mathematics Part-II textbook.
Integration as the Inverse of Differentiation
If \( \frac{d}{dx}F(x) = f(x) \), then \( \int f(x)\,dx = F(x) + C \) is the indefinite integral of f, and the family \( \{F + C : C \in \mathbb{R}\} \) contains every possible antiderivative of f (NCERT, p. 227). The arbitrary real number C is the constant of integration.
Functions with the same derivative differ by a constant, so the constant C is the only difference between two antiderivatives. Given \( \frac{dy}{dx} = f(x) \), we write \( y = \int f(x)\,dx \).
Standard Integrals
These come from the derivative table by reversing each row (NCERT, pp. 228–229). They are the building blocks used by every other method in this chapter.
\[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\ n \neq -1;\qquad \int dx = x + C \]
\[ \int \cos x\,dx = \sin x + C,\qquad \int \sin x\,dx = -\cos x + C \]
\[ \int \sec^2 x\,dx = \tan x + C,\qquad \int \operatorname{cosec}^2 x\,dx = -\cot x + C \]
\[ \int \sec x \tan x\,dx = \sec x + C,\qquad \int \operatorname{cosec} x \cot x\,dx = -\operatorname{cosec} x + C \]
\[ \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1} x + C\ \left(\text{also } -\cos^{-1} x + C\right),\qquad \int \frac{dx}{1+x^2} = \tan^{-1} x + C\ \left(\text{also } -\cot^{-1} x + C\right) \]
\[ \int e^x\,dx = e^x + C,\qquad \int a^x\,dx = \frac{a^x}{\log a} + C,\qquad \int \frac{1}{x}\,dx = \log|x| + C \]
For trig powers and trig products, first rewrite the integrand using identities such as \( \cos^2 x = \frac{1+\cos 2x}{2} \) and \( \sin x \cos y = \frac{1}{2}[\sin(x+y) + \sin(x-y)] \) (NCERT, p. 242).
Properties of Indefinite Integrals
Differentiation and integration are inverse operations, and the indefinite integral is linear (NCERT, pp. 230–231).
\[ \frac{d}{dx}\int f(x)\,dx = f(x),\qquad \int f'(x)\,dx = f(x) + C \]
\[ \int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx,\qquad \int k\,f(x)\,dx = k\int f(x)\,dx,\ k \in \mathbb{R} \]
Generalised to a finite combination of functions:
\[ \int [k_1 f_1(x) + k_2 f_2(x) + \dots + k_n f_n(x)]\,dx = k_1\int f_1(x)\,dx + k_2\int f_2(x)\,dx + \dots + k_n\int f_n(x)\,dx \]
Integration by Substitution
Change the variable from x to t by putting \( x = g(t) \). Then \( dx = g'(t)\,dt \), and the integral transforms as follows (NCERT, p. 236):
\[ \int f(x)\,dx = \int f(g(t))\,g'(t)\,dt \]
Four standard results obtained by substitution; the textbook says these are used later without reference (NCERT, pp. 237–238):
\[ \int \tan x\,dx = \log|\sec x| + C,\qquad \int \cot x\,dx = \log|\sin x| + C \]
\[ \int \sec x\,dx = \log|\sec x + \tan x| + C,\qquad \int \operatorname{cosec} x\,dx = \log|\operatorname{cosec} x – \cot x| + C \]
Integrals of Some Particular Functions
Six standard forms for expressions built from \( a^2 \) and \( x^2 \) (NCERT, pp. 243–246):
\[ \int \frac{dx}{x^2-a^2} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right| + C,\qquad \int \frac{dx}{a^2-x^2} = \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right| + C \]
\[ \int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C \]
\[ \int \frac{dx}{\sqrt{x^2-a^2}} = \log\left|x + \sqrt{x^2-a^2}\right| + C,\qquad \int \frac{dx}{\sqrt{x^2+a^2}} = \log\left|x + \sqrt{x^2+a^2}\right| + C \]
\[ \int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a} + C \]
For a quadratic denominator or a quadratic under a root, complete the square first (NCERT, pp. 246–247):
- With \( ax^2+bx+c = a\left[\left(x+\frac{b}{2a}\right)^2 + \left(\frac{c}{a}-\frac{b^2}{4a^2}\right)\right] \) and \( t = x + \frac{b}{2a} \), the integral \( \int \frac{dx}{ax^2+bx+c} \) reduces to \( \frac{1}{a}\int \frac{dt}{t^2 \pm k^2} \).
- The same completion reduces \( \int \frac{dx}{\sqrt{ax^2+bx+c}} \) to one of the square-root forms above.
- For a linear numerator, write \( px+q = A(2ax+b) + B \), find A and B by equating coefficients, then split the integral into a log part and a standard form.
Integration by Partial Fractions
For a proper rational function \( \frac{P(x)}{Q(x)} \) whose denominator factorises, decompose the integrand and integrate term by term. If the fraction is improper, divide first: \( \frac{P(x)}{Q(x)} = T(x) + \frac{P_1(x)}{Q(x)} \) (NCERT, p. 253). The decomposition forms are:
| Form of the rational function | Form of the partial fraction |
|---|---|
| \( \frac{px+q}{(x-a)(x-b)},\ a \neq b \) | \( \frac{A}{x-a} + \frac{B}{x-b} \) |
| \( \frac{px+q}{(x-a)^2} \) | \( \frac{A}{x-a} + \frac{B}{(x-a)^2} \) |
| \( \frac{px^2+qx+r}{(x-a)(x-b)(x-c)} \) | \( \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} \) |
| \( \frac{px^2+qx+r}{(x-a)^2(x-b)} \) | \( \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b} \) |
| \( \frac{px^2+qx+r}{(x-a)(x^2+bx+c)} \) where \( x^2+bx+c \) cannot be factorised further | \( \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c} \) |
Integration by Parts
For two differentiable functions f and g (NCERT, p. 260):
\[ \int f(x)g(x)\,dx = f(x)\int g(x)\,dx – \int\left[f'(x)\int g(x)\,dx\right]dx \]
In words: first function × integral of the second − integral of (derivative of the first × integral of the second). Choose the first function so the new integral is simpler — take a polynomial as the first function; take \( \log x \) or an inverse-trigonometric function as the first function when one is present.
Special form. If the integrand is \( e^x[f(x) + f'(x)] \), the result simplifies directly (NCERT, p. 263):
\[ \int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C \]
Integrals of Some More Types
Three square-root integrals obtained by integrating by parts with 1 as the second function (NCERT, p. 265):
\[ \int \sqrt{x^2-a^2}\,dx = \frac{x}{2}\sqrt{x^2-a^2} – \frac{a^2}{2}\log\left|x + \sqrt{x^2-a^2}\right| + C \]
\[ \int \sqrt{x^2+a^2}\,dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\log\left|x + \sqrt{x^2+a^2}\right| + C \]
\[ \int \sqrt{a^2-x^2}\,dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C \]
Definite Integrals
A definite integral \( \int_a^b f(x)\,dx \) has a unique value; a is the lower limit and b the upper limit. By the second fundamental theorem of integral calculus (NCERT, pp. 267–268):
\[ \int_a^b f(x)\,dx = [F(x)]_a^b = F(b) – F(a) \]
Steps: find an antiderivative F of f, then evaluate \( F(b) – F(a) \). The constant C is not needed because it cancels between the two limits. The function f must be well defined and continuous on \( [a, b] \).
The area function \( A(x) = \int_a^x f(t)\,dt \) satisfies \( A'(x) = f(x) \) on \( [a, b] \) (first fundamental theorem, NCERT, p. 268).
Properties of Definite Integrals
The seven properties P0–P7 of the textbook speed up evaluation, especially for symmetric limits and trigonometric integrands (NCERT, pp. 273–274):
| Property | Formula | What it lets you do |
|---|---|---|
| P0 | \( \int_a^b f(x)\,dx = \int_a^b f(t)\,dt \) | Rename the variable — the variable of integration is a dummy. |
| P1 | \( \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx;\quad \int_a^a f(x)\,dx = 0 \) | Reversing the limits changes the sign. |
| P2 | \( \int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx \) | Split the interval at a point — needed for \( |f(x)| \) integrals. |
| P3 | \( \int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx \) | Replace x by \( a+b-x \); simplifies many trig integrands. |
| P4 | \( \int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx \) | Special case of P3 for limits 0 to a; the workhorse of symmetric trig integrals. |
| P5 | \( \int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a-x)\,dx \) | Split a 0 to 2a interval into two halves. |
| P6 | \( \int_0^{2a} f(x)\,dx = 2\int_0^a f(x)\,dx \) if \( f(2a-x) = f(x) \), and 0 if \( f(2a-x) = -f(x) \) | Exploit symmetry about x = a after testing \( f(2a-x) \). |
| P7 | \( \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \) if f is even \( (f(-x) = f(x)) \); 0 if f is odd \( (f(-x) = -f(x)) \) | Symmetric limits: check even/odd before integrating. |
What Each Symbol Means
Units do not apply in pure mathematics, so the third column below states the nature of each quantity instead.
| Symbol | What it means | Unit / nature |
|---|---|---|
| \( \int f(x)\,dx \) | Indefinite integral of f with respect to x | A family of functions |
| \( f(x) \) | Integrand — the function being integrated | A real-valued function |
| \( dx \) | Differential of x; integration is carried out with respect to x | An infinitesimal change in x |
| \( x \) | Variable of integration | A real number |
| \( F(x) \) | Antiderivative (primitive) of f, so that \( F'(x) = f(x) \) | A real-valued function |
| \( C \) | Constant of integration — an arbitrary real number | A real number (dimensionless) |
| \( n \) | Exponent in the power rule | A real number, \( n \neq -1 \) |
| \( k, k_1, \dots, k_n \) | Constant multipliers in the linearity property | Real numbers (dimensionless) |
| \( a \) in \( x^2-a^2, a^2-x^2, x^2+a^2 \) | Positive constant defining the standard form | A positive real number |
| \( a, b \) in \( \int_a^b \) | Lower and upper limits of the definite integral | Real numbers (endpoints of the interval) |
| \( g(t) \) | Substitution function: \( x = g(t) \) | A differentiable function |
| \( f, g \) in integration by parts | The two factors of the product being integrated | Differentiable functions |
| \( P(x), Q(x) \) | Numerator and denominator polynomials of a rational function | Polynomials, \( Q(x) \neq 0 \) |
When to Use Each Formula
Match the shape of your integrand to the row below; the condition column is the check to run before applying the formula.
| Situation | Formula / method | Condition to check |
|---|---|---|
| Integral of a polynomial or sum of powers | Power rule with property V (split term by term) | Exponent \( n \neq -1 \) for each term |
| Integrand contains \( \frac{1}{x} \) | \( \log|x| + C \) form | This is the \( n = -1 \) case of the power rule |
| Integrand contains \( e^x \) or \( a^x \) | Exponential forms | \( a \gt 0 \) |
| Direct trig integrands: sin, cos, sec², cosec², sec·tan, cosec·cot | Standard integral table | The integrand matches the derivative table exactly |
| Integrand has tan x, cot x, sec x, cosec x | Log forms from substitution | Rewrite in sin/cos first if the form is not obvious |
| A function and its derivative both appear (e.g. \( 2x \) with \( x^2+1 \), cos x with sin x) | Integration by substitution, \( x = g(t) \) | Replace dx by \( g'(t)\,dt \) completely |
| Denominator is \( x^2 \pm a^2 \), \( x^2+a^2 \), or a square root of one of these | The six particular-function forms | Identify \( a^2 \) first; the signs decide which form fits |
| Denominator is \( ax^2+bx+c \) or \( \sqrt{ax^2+bx+c} \) | Complete the square, then use a particular form | Put \( t = x + \frac{b}{2a} \); check the sign of the leftover constant |
| Numerator is linear (px + q) over a quadratic | Write \( px+q = A(2ax+b) + B \), split the integral | Find A and B by equating coefficients |
| Rational function with a factorable denominator | Partial fractions | Proper fraction; if improper, divide first; pick the right row of the decomposition table |
| Product of two unlike functions (x·sin x, x·log x, x·eˣ) | Integration by parts | Choose the first function so the next integral is simpler |
| Integrand is \( e^x \) times (a function + its derivative) | Special form \( \int e^x[f(x)+f'(x)]dx = e^x f(x) + C \) | Spot f by differentiating candidate functions |
| Integrand has \( \sqrt{x^2 \pm a^2} \) or \( \sqrt{a^2-x^2} \) | Three square-root integrals (by parts) | Complete the square inside the root if needed |
| Any definite integral with a known antiderivative | Second fundamental theorem: \( F(b) – F(a) \) | f continuous on \( [a, b] \); no + C needed |
| Definite integral that needs substitution | Substitution, then change the limits | New limits belong to the new variable |
| Symmetric limits \( [-a, a] \) | Property P7 | Test \( f(-x) \): even gives \( 2\int_0^a \), odd gives 0 |
| Limits \( [0, 2a] \) or \( [0, a] \) with a special symmetry | Properties P6, P4, P3 | Check \( f(2a-x) \) or \( f(a-x) \) before applying |
Worked Examples
Three original examples showing formula selection, substitution, and the final check. For the textbook’s own practice sets and the full set of Class 12 chapter formula sheets, see the Maths formulas index.
Example 1: Integrate \(4x^3 – 3\sin x + 2e^x\)
Step 1: Split the integral term by term using property V (linearity of the indefinite integral).
\[ \int (4x^3 – 3\sin x + 2e^x)\,dx = 4\int x^3\,dx – 3\int \sin x\,dx + 2\int e^x\,dx \]
Step 2: Apply the power rule to \( x^3 \), the sine rule to \( \sin x \), and the exponential rule to \( e^x \).
Remember \( \int \sin x\,dx = -\cos x + C \).
\[ = 4\cdot\frac{x^4}{4} – 3(-\cos x) + 2e^x + C = x^4 + 3\cos x + 2e^x + C \]
Final answer: \( \int (4x^3 – 3\sin x + 2e^x)\,dx = x^4 + 3\cos x + 2e^x + C \). Check by differentiating: \( \frac{d}{dx}(x^4 + 3\cos x + 2e^x) = 4x^3 – 3\sin x + 2e^x \).
Example 2: Find the antiderivative F of \(f(x) = 6x^2 – 8x\) with \(F(0) = 5\)
Step 1: Integrate f(x) and write the general antiderivative with the constant C.
\[ F(x) = \int (6x^2 – 8x)\,dx = 6\cdot\frac{x^3}{3} – 8\cdot\frac{x^2}{2} + C = 2x^3 – 4x^2 + C \]
Step 2: Use the given condition \( F(0) = 5 \) to fix the value of C.
\[ F(0) = 2(0)^3 – 4(0)^2 + C = C = 5 \]
Final answer: \( F(x) = 2x^3 – 4x^2 + 5 \). This is the unique antiderivative of f that satisfies the condition.
Example 3: Evaluate \(\int_1^2 (3x^2 + 2x)\,dx\)
Step 1: Find the antiderivative F.
For a definite integral the constant C is not needed — it cancels between the limits.
\[ F(x) = \int (3x^2 + 2x)\,dx = x^3 + x^2 \]
Step 2: Apply the second fundamental theorem: substitute the upper limit, subtract the value at the lower limit.
\[ \int_1^2 (3x^2 + 2x)\,dx = [x^3 + x^2]_1^2 = (8 + 4) – (1 + 1) = 10 \]
Final answer: \( \int_1^2 (3x^2 + 2x)\,dx = 10 \).
Common Mistakes to Avoid
These are the errors that appear while applying this chapter’s formulas — each comes with a check you can run on your own answer.
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| Using the power rule \( \int x^n dx = \frac{x^{n+1}}{n+1} \) for \( n = -1 \) | When \( n = -1 \), the formula is \( \int \frac{1}{x}\,dx = \log|x| + C \) | Differentiate your answer: \( \frac{d}{dx}\log|x| = \frac{1}{x} \), but \( \frac{d}{dx}\frac{x^0}{0} \) is undefined. |
| Dropping + C from an indefinite integral | Every indefinite integral ends with + C; the constant disappears only in a definite integral | Look at the integral sign: no limits → the answer must contain C. |
| In a definite integral, substituting \( x = g(t) \) but keeping the original limits | Either change the limits with the variable, or resubstitute back to x before using the old limits | Test with a simple case like \( \int_0^1 2x\,dx \): both routes must give 1. |
| Wrong choice of first function in integration by parts | Take a polynomial as the first function; take \( \log x \) or \( \sin^{-1}x \) as the first function when present | The new integral must be simpler than the old one — if it gets harder, swap the functions. |
| Applying P7 to a function before testing even/odd | Compute \( f(-x) \) explicitly: even gives \( 2\int_0^a f(x)dx \), odd gives 0 | Quick test: \( x^2 \) is even, \( x^3 \) is odd; write out \( f(-x) \) for the actual integrand. |
Frequently Asked Questions
Why is + C added to every indefinite integral?
Because the derivative of any constant is zero. If \( F'(x) = f(x) \), then \( (F + C)'(x) = f(x) \) for every real number C, so \( \int f(x)\,dx = F(x) + C \) represents the whole family of antiderivatives. In a definite integral the constant cancels: \( [F(x) + C]_a^b = F(b) – F(a) \).
How do I choose between substitution, partial fractions, and integration by parts?
Look at the shape of the integrand:
- A rational function whose denominator factorises → partial fractions.
- A product of two unlike functions (\( x\sin x \), \( x\log x \), \( xe^x \)) → integration by parts.
- A function whose derivative also appears in the integrand (\( x^2 \) with \( x^3+1 \), cos x with sin x) → substitution.
- \( e^x \) times (a function + its derivative) → the special form \( \int e^x[f(x)+f'(x)]dx = e^x f(x) + C \).
Do the limits change when I substitute in a definite integral?
Yes. There are two correct routes: (1) substitute, convert the limits to the new variable, integrate, and stop; or (2) integrate in the new variable, substitute back to x, and then use the original limits. Mixing the two — new integrand with old limits — gives a wrong value.
What is the difference between an indefinite and a definite integral?
An indefinite integral \( \int f(x)\,dx \) is a family of functions \( F(x) + C \). A definite integral \( \int_a^b f(x)\,dx \) is a single number \( F(b) – F(a) \), equal to the area under \( y = f(x) \) between \( x = a \) and \( x = b \) when \( f(x) \geq 0 \). The second fundamental theorem connects the two.
Reference: NCERT Class 12 Mathematics textbook, chapter Integrals.
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