JEE Mathematics Integrals

Indefinite Integration Formulas

Definition:
If \( f \) & \( g \) are functions of \( x \) such that \( g'(x) = f(x) \) then, \( \int f(x) dx = g(x) + c \iff \frac{d}{dx} \{g(x)+c\} = f(x) \), where \( c \) is called the constant of integration.

Standard Formula:
(i) \( \int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n+1)} + c, n \neq -1 \)
(ii) \( \int \frac{dx}{ax + b} = \frac{1}{a} \ln (ax + b) + c \)
(iii) \( \int e^{ax+b} dx = \frac{1}{a} e^{ax+b} + c \)
(iv) \( \int a^{px+q} dx = \frac{1}{p} \frac{a^{px+q}}{\ln a} + c ; a > 0 \)
(v) \( \int \sin(ax + b) dx = -\frac{1}{a} \cos (ax + b) + c \)
(vi) \( \int \cos(ax + b) dx = \frac{1}{a} \sin (ax + b) + c \)
(vii) \( \int \tan(ax + b) dx = \frac{1}{a} \ln \sec (ax + b) + c \)
(viii) \( \int \cot(ax + b) dx = \frac{1}{a} \ln \sin (ax + b) + c \)
(ix) \( \int \sec^2(ax + b) dx = \frac{1}{a} \tan (ax + b) + c \)
(x) \( \int \text{cosec}^2(ax + b) dx = -\frac{1}{a} \cot (ax + b) + c \)
(xiii) \( \int \sec x dx = \ln (\sec x + \tan x) + c \) OR \( \ln \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) + c \)
(xiv) \( \int \text{cosec } x dx = \ln (\text{cosec } x - \cot x) + c \) OR \( \ln \tan \frac{x}{2} + c \)
(xv) \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \frac{x}{a} + c \)
(xvi) \( \int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + c \)
(xvii) \( \int \frac{dx}{|x| \sqrt{x^2 - a^2}} = \frac{1}{a} \sec^{-1} \frac{x}{a} + c \)
(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) \( \int \sqrt{x^2 - a^2} dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left( \frac{x + \sqrt{x^2 - a^2}}{a} \right) + c \)

Integration by Part :
\( \int (f(x) g(x)) dx = f(x) \int g(x) dx - \int \left( \frac{d}{dx} (f(x)) \int (g(x)) dx \right) dx \)

Definite Integration Formulas

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. \( \int_{-a}^a f(x) dx = \begin{cases} 2 \int_0^a f(x) dx & f(-x) = f(x) \\ 0 & f(-x) = -f(x) \end{cases} \)
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. \( \int_0^{2a} f(x) dx = \int_0^a (f(x) + f(2a-x)) dx = \begin{cases} 2 \int_0^a f(x) dx & f(2a-x) = f(x) \\ 0 & f(2a-x) = -f(x) \end{cases} \)
8. If \( f(x) \) is a periodic function with period T, then \( \int_0^{nT} f(x) dx = n \int_0^T f(x) dx, n \in Z \).
\( \int_a^{a+nT} f(x) dx = n \int_0^T f(x) dx, n \in z, a \in R \).
\( \int_{mT}^{nT} f(x) dx = (n - m) \int_0^T f(x) dx, m, n \in z \).
\( \int_{nT}^{a+nT} f(x) dx = \int_0^a f(x) dx, n \in z, a \in R \).
\( \int_{a+nT}^{b+nT} f(x) dx = \int_a^b f(x) dx, n \in z, a, b \in R \).

Leibnitz Theorem :
If \( F(x) = \int_{g(x)}^{h(x)} f(t) dt \), then \( \frac{dF(x)}{dx} = h'(x) f(h(x)) - g'(x) f(g(x)) \)