# Important Integration Formulas for JEE Main and Advanced

In this section, students will learn the main indefinite and definite integration formulas as well as some main properties of integration. In general, integration is the reverse operation of differentiation. It is also called antiderivative. The formulas provided here will help students to easily remember them for the exam and score higher marks in the exams.

## Integration Formulas

### Indefinite Integration:

1. If f and g are functions of x such that g’(x) = f(x) then,

∫ f(x)dx = g(x)+c  ⇔ $\frac{d}{dx}\left \{g(x)+c \right \} = f(x)$ , where c is called the constant of integration.

1. Standard formula:

(a)  ∫xn dx = $\frac{x^{n+1}}{n+1}+c$ , n ≠ -1

(b) $\int \frac{1}{x}\: dx = \log_{e}\left | x \right |+c$

(c) ∫ex dx = ex+c

(d) ∫ax dx = $\frac{a^{x}}{\log_{e}a}+c$

(e) ∫ sin x dx  = -cos x + c

(f) ∫ cos x dx  = sin x + c

(g) ∫ sec2 x dx  = tan x + c

(h) ∫ cosec2 x dx  = -cot x + c

(h) ∫ sec x tan x dx = sec x + c

(i) ∫ cosec x cot x dx = – cosec x + c

(j) ∫ cot x dx = $\log \left | \sin x \right |+c$

(k) ∫ tan x dx = $-\log \left | \\cos \: x \right |+c$

(l) ∫ sec x dx = log $\left | \\sec \: x + tan \: x\right |+c$

(m) ∫ cosec x dx = log $\left | \\cosec \: x – \cot \: x\right |+c$

(n) $\int \frac{1}{\sqrt{a^{2}-x^{2}}} \; dx = \sin ^{-1}(\frac{x}{a})+c$

(o) $\int -\frac{1}{\sqrt{a^{2}-x^{2}}} \; dx = \cos ^{-1}(\frac{x}{a})+c$

(p) $\int \frac{1}{{a^{2}+x^{2}}} \; dx = \frac{1}{a}\tan ^{-1}(\frac{x}{a})+c$

(q) $\int -\frac{1}{{a^{2}+x^{2}}} \; dx = \frac{1}{a}\cot ^{-1}(\frac{x}{a})+c$

(r) $\int \frac{1}{x\sqrt{x^{2}-a^{2}}}\; dx = \frac{1}{a}\sec ^{-1}(\frac{x}{a})+c$

(s) $\int -\frac{1}{x\sqrt{x^{2}-a^{2}}}\; dx = \frac{1}{a}\: cosec ^{-1}\left ( \frac{x}{a} \right )+c$

If in place of x we have (ax+b), then the same formula is applicable but we must divide by coefficient of x or derivative of (ax+b) i.e. a.

(i) ∫ (ax+b)n dx = $\frac{(ax+b)^{n+1}}{a(n+1)} \: + c$ , n ≠ -1

(ii) $\int \frac{dx}{ax+b} = \frac{1}{a}$ ln(ax+b) +c

(iii) ∫ eax+b dx = (1/a) eax+b+c

(iv) ∫ apx+q dx = $\frac{1}{p} \frac{a^{px+q}}{\ln a} \: +c$ , a>0

(v)  ∫ sin (ax+b)dx = -(1/a) cos (ax+b) + c

(vi) ∫ cos (ax+b) dx = (1/a) sin (ax+b) + c

(vii)  ∫ tan (ax+b)dx = (1/a) ln sec (ax+b) + c

(viii) ∫ cot (ax+b)dx = (1/a) ln sin (ax+b) + c

(ix) ∫ sec2 (ax+b)dx = (1/a) tan (ax+b) + c

(x) ∫ cosec2 (ax+b) dx = -(1/a) cot (ax+b) + c

(xi) ∫ sec (ax+b) ⋅ tan (ax+b) dx = (1/a) sec (ax+b) + c

(xii) ∫ cosec (ax+b) ⋅ cot (ax+b) dx = -(1/a) cosec (ax+b) + c

(xiii) ∫ sec x dx = ln (sec x + tan x) + c  or  $\ln tan \left ( \frac{\pi }{4} +\frac{x}{2}\right )$

(xiv) ∫ cosec x dx = ln (cosec x – cot x) + c  or  ln tan (x/2) +c  or –ln (cosec x + cot x)+ c

(xv)  $\int \frac{dx}{\sqrt{a^{2}-x^{2}}} = \sin ^{-1} \frac{x}{a} +c$

(xvi) $\int \frac{dx}{a^{2}+x^{2}} = \frac{1}{2} \tan ^{-1} \frac{x}{a} +c$

(xvii) $\int \frac{dx}{\left | x \right |\sqrt{x^{2}-a^{2}}} = \frac{1}{a} \sec ^{-1}\frac{x}{a}$

(xviii) $\int \frac{dx}{\sqrt{x^{2}+a^{2}}} = \ln [x+\sqrt{x^{2}+a^{2}}]+c$

(xix) $\int \frac{dx}{\sqrt{x^{2}-a^{2}}} = \ln [x+\sqrt{x^{2}-a^{2}}]+c$

(xx) $\int \frac{dx}{{a^{2}-x^{2}}} = \frac{1}{2a}\ln \left | \frac{a+x}{a-x} \right |+c$

(xxi) $\int \frac{dx}{{x^{2}-a^{2}}} = \frac{1}{2a}\ln \left | \frac{x-a}{x+a} \right |+c$

(xxii) $\int \sqrt{a^{2}-x^{2}} \: dx = \frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}\sin ^{-1}\frac{x}{a}+c$

(xxiii) $\int \sqrt{x^{2}+a^{2}} \: dx = \frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}\ln \left ( \frac{x+\sqrt{x^{2}+a^{2}}}{a} \right )+c$

(xxiv) $\int \sqrt{x^{2}-a^{2}} \: dx = \frac{x}{2}\sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2}\ln \left ( \frac{x+\sqrt{x^{2}-a^{2}}}{a} \right )+c$

(xxv) $\int e^{ax}.\sin bx \: dx = \frac{e^{ax}}{a^{2}+b^{2}}\left ( a\, \sin bx-b\cos bx \right )+c$

(xxvi) $\int e^{ax}.\cos bx \: dx = \frac{e^{ax}}{a^{2}+b^{2}}\left ( a\, \cos bx+b\: \sin bx \right )+c$

1. Theorems on integration:

(i) ∫ c f(x) dx = c ∫ f(x) dx

(ii) $\int (f(x)\pm g(x))\: dx = \int f(x)\: dx\pm g(x)\: dx$

(iii) ∫ f(x) dx =  g(x)+c ⇒ ∫ f(ax+b) dx = $\frac{g(ax+b)}{a}+c$

1. Integration by substitutions:

If we substitute f(x) = t, then f’(x) dx = dt

1. Integration by part:

∫( f(x) g(x)) dx = f(x) ∫g(x) dx – $\int\left ( \frac{d}{dx}(f(x))\int (g(x))dx\right ) dx$

(i) when you find integral ∫g(x) dx then it will not contain an arbitrary constant.

(ii) ∫g(x) dx should be taken as the same in both terms.

(iii) the choice of f(x) and g(x) is decided by ILATE rule.

1. Integration of type

$\int \frac{dx}{ax^{2}+bx+c}$  , $\int \frac{dx}{\sqrt{ax^{2}+bx+c}}$  , $\int \sqrt{ax^{2}+bx+c}\; dx$

Make the substitution $x+\frac{b}{2a} = t$

1. Integration of type

(1) $\int \frac{px+q}{ax^{2}+bx+c}\: dx$$\int \frac{px+q}{\sqrt{ax^{2}+bx+c}}\: dx$ , $\int (px+q)\sqrt{ax^{2}+bx+c}\; dx$

Make the substitution $x+\frac{b}{2a} = t$ , then split the integral as sum of two integrals one containing the linear term and the other containing constant term.

8. Integration of trigonometric functions

(1) $\int \frac{dx}{a+b\: \sin ^{2}x}$  or $\int \frac{dx}{a+b\: \cos ^{2}x}$  or $\int \frac{dx}{a \sin ^{2}x+b \sin x\cos x+c\cos ^{2}x}$  put tan x = t.

(2) $\int \frac{dx}{a+b\sin x}$  or $\int \frac{dx}{a+b\cos x}$  or

$\int \frac{dx}{a+b\sin x+c \cos x}$  put tan (x/2) = t.

(3) $\int \frac{a\cos x+ b\sin x+c}{l\cos x+ m\sin x+n}dx$

Express Nr = $A(Dr)+B\frac{d}{dx}(Dr)+c$  and proceed.

1. Integration of type ∫sin mx. cos nx dx

Case 1. If m and n are even natural numbers then express sin mx cos nx in the terms of sines and cosines of multiples of x by using trigonometric results or De’ Moivere’s theorem.

Case 2. If m is an odd natural number then put cos x = t.

If n is an odd natural number then put sin x = t.

If both m and n are odd natural numbers then put either sin x = t or cos x = t.

Case 3. When m+n is a negative even integer then put tan x = t.

1. Integration of type  $\int \frac{x^{2}\pm 1}{x^{4}+Kx^{2}+1}\: dx$ where K is any constant.

Divide numerator and denominator by x2 and put $x\pm \frac{1}{x} = t$

1. Integration of type $\int \frac{dx}{(ax+b)\sqrt{px+q}}$ or $\int \frac{dx}{(ax^{2}+bx+c)\sqrt{px+q}}$ :

Put px+q = t2

1. Integration of type

(i) $\int \frac{dx}{(ax+b)\sqrt{px^{2}+qx+r}}$ : put ax+b = $\frac{1}{t}$

(ii) $\int \frac{dx}{(ax^{2}+b)\sqrt{px^{2}+q}}$ : put x = $\frac{1}{t}$

1. Integration of type

(i) $\int \sqrt{\frac{x-\alpha }{\beta -x}} \: dx$ or $\int \sqrt{(x-\alpha )(\beta -x)}$ : put x = $\alpha \cos ^{2}\theta +\beta \sin ^{2}\theta$

(ii) $\int \sqrt{\frac{x-\alpha }{x -\beta }} \: dx$ or $\int \sqrt{(x-\alpha )(x-\beta )}$ : put x = $\alpha \sec ^{2}\theta -\beta \tan ^{2}\theta$

(iii) $\int \frac{dx}{\sqrt{(x-\alpha )(x-\beta )}}$ : put x-α = t2 or x-β = t2

1. Reduction formula of ∫ tann x dx, ∫ cotn x dx, ∫ secn x dx, ∫ cosecn x dx:

(i) If In = ∫ tann x dx, then In = $\frac{tan^{n-1}x}{n-1}-I_{n-2}$

(ii) If In = ∫ cotn x dx, then  In = $-\frac{\cot ^{n-1}x}{n-1}-I_{n-2}$

(iii) If In = ∫ secn x dx, then In = $\frac{\\tan x \: \sec ^{n-2}x}{n-1}+\frac{n-2}{n-1}\: I_{n-2}$

(iv) If In = ∫ cosecn x, then  In = $\frac{\cot x \; cosec^{n-2}x}{n-1}+\frac{n-2}{n-1}I_{n-2}$

### Definite Integration:

Properties of definite integral:

(1) $\int_{a}^{b} f(x)dx = \int_{a}^{b}f(t)dt$

(2) $\int_{a}^{b} f(x)dx = -\int_{b}^{a}f(x)dx$

(3) $\int_{a}^{b} f(x)dx = \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx$

(4) $\int_{-a}^{a} f(x)dx = \int_{0}^{a} (f(x)+f(-x))dx$ = $\left\{\begin{matrix} 2\int_{0}^{a}f(x)dx\: ,&f(-x) = f(x) \\ 0\: ,& f(-x) = -f(x) \end{matrix}\right.$

(5) $\int_{a}^{b} f(x)dx = \int_{a}^{b}f(a+b-x)dx$

(6) $\int_{0}^{a} f(x)dx = \int_{0}^{a}f(a-x)dx$

(7) $\int_{0}^{2a} f(x)dx = \int_{0}^{a}(f(x)+f(2a-x))dx$ = $\left\{\begin{matrix} 2\int_{0}^{a}f(x)dx\: ,&f(2a-x) = f(x) \\ 0\: ,& f(2a-x) = -f(x) \end{matrix}\right.$

(8) If f(x) is a periodic function with period T, then

(i) $\int_{0}^{nT}f(x)dx = n\int_{0}^{T}f(x)dx ,$, n∈Z .

(ii) $\int_{0}^{a+nT}f(x)dx = n\int_{0}^{T}f(x)dx ,$, n∈Z, a∈R .

(iii) $\int_{mt}^{nT}f(x)dx = (n-m)\int_{0}^{T}f(x)dx ,$, m, n∈Z.

(iv) $\int_{nt}^{a+nT}f(x)dx = \int_{0}^{a}f(x)dx ,$, n∈Z, a∈R .

(9) If $\psi (x)\leq f(x)\leq \phi (x) for\ a\leq x\leq b\ then \int_{a}^{b}\psi (x)dx \leq \int_{a}^{b}f(x)dx\leq \int_{a}^{b}\phi (x)$

(10) If m≤ f(x) ≤ M for a≤x≤b, then m(b-a) ≤ $\int_{a}^{b}f(x) dx\leq M(b-a)$

(11) $\left | \int_{a}^{b}f(x)dx \right |\leq \int_{a}^{b}\left | f(x) \right |dx$

(12) If f(x) ≥0 on [a,b] then $\int_{a}^{b}f(x)dx\geq 0$

### Leibnitz Theorem:

If F(x) = $\int_{g(x)}^{h(x)}f(t)dt$, then $\frac{dF(x)}{dx} =$ h’(x)f(h(x))-g’(x)f(g(x))

Definite integrals as a limit of sum:

(i) $\int_{a}^{b}f(x)dx=\lim_{n\to\infty }\sum_{r=0}^{n-1}h\: f(a+rh) = \lim_{n\to\infty }\sum_{r=0}^{n-1}\left ( \frac{b-a}{n} \right )f\left ( a+\frac{(b-a)r}{n} \right )$