# Integration

UPI ID:- achalup41-1@oksbi

Contact Email ID:- achalup41@gmail.com

### Integration of the composite function

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C, n\neq 1$$

$$\int \frac{1}{x}dx=log|x|+C$$

$$\int e^{x}dx=e^{x}+C$$

'C' is the Integration constant.

### Integration formula of the Trigonometric function

$$\int sin x\hspace{2mm} dx=-cos x+C$$

$$\int cos x\hspace{2mm} dx=sin x+C$$

$$\int sec^{2}x\hspace{2mm} dx =tan x+C$$

$$\int cosec^{2}x\hspace{2mm} dx=-cotx+C$$

$$\int secx \hspace{1mm} tanx\hspace{2mm}dx = secx+C$$

$$\int cosecx \hspace{1mm} cotx\hspace{2mm}dx = -cosecx+C$$

$$\int tanx \hspace{2mm}dx = log(secx)+C$$

$$\int tanx \space dx = -log(cosx)+C$$

$$\int cotx \hspace{2mm}dx = log(sinx)+C$$

$$\int cotx \space dx = -log(cosecx)+C$$

$$\int cosecx \hspace{2mm}dx = log(cosecx - cotx)+C$$

$$\int cosecx \hspace{2mm}dx = log(tan\frac{x}{2})+C$$

$$\int secx \hspace{2mm}dx = log(secx+tanx)+C$$

$$\int secx \hspace{2mm}dx = log\hspace{1mm}tan\left ( \frac{π}{4}+\frac{x}{2} \right )+C$$

'C' is the Integration constant.

### Integration into Inverse Trigonometric Functions

$$\int \frac{1}{\sqrt{1-x^{2}}}\hspace{2mm}dx=sin^{-1}x+C$$

$$\int -\frac{1}{\sqrt{1-x^{2}}}\hspace{2mm}dx=cos^{-1}x+C$$

$$\int \frac{1}{1+x^{2}}\hspace{2mm}dx=tan^{-1}x+C$$

$$\int -\frac{1}{1+x^{2}}\hspace{2mm}dx=cot^{-1}x+C$$

$$\int \frac{1}{x\sqrt{x^{2}-1}}\hspace{2mm}dx=sec^{-1}x+C$$

$$\int -\frac{1}{x\sqrt{x^{2}-1}}\hspace{2mm}dx=-cosec^{-1}x+C$$

$$\int \frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}tan^{-1}\left ( \frac{x}{a} \right )+C$$

$$\int \frac{1}{\sqrt{a^{2}-x^{2}}}dx=sin^{-1}\left ( \frac{x}{a} \right )+C$$

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

'C' is the Integration constant.

### Integration by substitution

$$\int \left ( ax+b \right )^{n}dx = \frac{1}{a}\frac{(ax+b)^{n+1}}{n+1}, When\hspace{1mm} n\neq -1+C$$

$$\int\frac{1}{ax+b}dx=\frac{1}{a}log(ax+b)+C$$

$$\int e^{ax+b}dx=\frac{1}{a}e^{ax+b}+C$$

$$\int e^{ax+b}dx=\frac{1}{a}\frac{e^{ax+b}}{log_ee}+C$$

$$\int sin(ax+b)dx=-\frac{1}{a}cos(ax+b)+C$$

$$\int cos(ax+b)dx= \frac{1}{a}sin(ax+b)+C$$

$$\int sec^{2}(ax+b)dx= \frac{1}{a}tan(ax+b)+C$$

$$\int cosec^{2}(ax+b)dx= -\frac{1}{a}cot(ax+b)+C$$

$$\int sec(ax+b)tan(ax+b)dx= \frac{1}{a}sec(ax+b)+C$$
$$\int cosec(ax+b)cot(ax+b)dx= -\frac{1}{a}cosec(ax+b)+C$$

'C' is the Integration constant.

### Few Special integrations

$$\int\frac{1}{\sqrt{x^2+a^2}}dx =log(x+\sqrt{x^{2}+a^{2}})+C$$

$$\int\frac{1}{\sqrt{x^2-a^2}}dx =log(x+\sqrt{x^{2}-a^{2}})+C$$

$$\int\frac{1}{\sqrt{1+x^2}}dx =log(x+\sqrt{1+x^{2}})+C$$

$$\int\frac{1}{\sqrt{x^2-1}}dx =log(x+\sqrt{x^{2}-1})+C$$

$$\int\frac{1}{x^{2}-a^{2}}dx=\frac{1}{2a}log\left|\frac{x-a}{x+a}\right |+C, when\hspace{1mm} x> a$$

$$\int\frac{1}{a^{2}-x^{2}}dx=\frac{1}{2a}log\left|\frac{a+x}{a-x}\right |+C, when\hspace{1mm} x < a$$

$$\int a^{x}\hspace{2mm}dx=\frac{a^{x}}{loga}+C$$

$$\int \frac{1}{ax^{2}+bx+c}dx=\frac{1}{\sqrt{b^{2}-4ac}}log\frac{2ax+b-\sqrt{b^{2}-4ac}}{2ax+b+\sqrt{b^{2}-4ac}}+C, When\hspace{1mm}b^{2}-4ac\hspace{1mm} is\hspace{1mm} positive$$
$$\int \frac{1}{ax^{2}+bx+c}dx=\frac{1}{\sqrt{b^{2}-4ac}}log\frac{2ax+b-\sqrt{b^{2}-4ac}}{2ax+b+\sqrt{b^{2}-4ac}}+C, When\hspace{1mm}b^{2}-4ac\hspace{1mm} is\hspace{1mm} positive$$
$$\int \frac{1}{ax^{2}+bx+c}dx=\frac{2}{\sqrt{4ac-b^{2}}}tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^{2}}}+C, When\hspace{1mm}b^{2}-4ac\hspace{1mm} is\hspace{1mm} negative$$
$$\int \frac{px+q}{\sqrt{ax^{2}+bx+c}}dx= \frac{p}{a}\sqrt{ax^{2}+bx+c}+\left ( q-\frac{bp}{2a} \right )\int \frac{1}{\sqrt{ax^{2}+bx+c}}dx+C$$
$$\int\frac{1}{a\hspace{1mm}sinx+b\hspace{1mm}cosx}dx=\frac{1}{\sqrt{a^{2}+b^{2}}}log\left [ tan\left\{\frac{x}{2}+\frac{1}{2}tan^{-1}\frac{b}{a} \right\} \right ]+C$$

'C' is the Integration constant.

### Integration of two functions or Integration by parts

$$\int f_{1}(x)f_{2}(x)\hspace{2mm}dx=f_{1}(x)\int F_{2}(x)dx-\int \left [ \left\{ \frac{d}{dx}f_{1}(x) \right\}.\int F_{2}(x)dx \right ]dx+C$$

$$\int e^{ax}cos \hspace{1mm}bx \hspace{2mm}dx= \frac{e^{ax}}{a^{2}+b^{2}}(a\hspace{1mm}cos\hspace{1mm}bx+b\hspace{1mm}sin\hspace{1mm}bx)+C$$

$$\int e^{ax}cos \hspace{1mm}bx \hspace{2mm}dx= \frac{e^{ax}}{\sqrt{a^{2}+b^{2}}}cos \left[ bx - tan^{-1}(\frac{b}{a}) \right ]+C$$

$$\int e^{ax}sin \hspace{1mm}bx \hspace{2mm}dx= \frac{e^{ax}}{a^{2}+b^{2}}(a\hspace{1mm}sin\hspace{1mm}bx-b\hspace{1mm}cos\hspace{1mm}bx)+C$$

$$\int e^{ax}sin \hspace{1mm}bx \hspace{2mm}dx= \frac{e^{ax}}{\sqrt{a^{2}+b^{2}}}sin \left[ bx - tan^{-1}(\frac{b}{a}) \right ]+C$$

$$\int \sqrt{a^{2}-x^{2}}dx=\frac{1}{2}\left [ x\sqrt{a^{2}-x^{2}}+a^{2}sin^{-1}(\frac{x}{a}) \right ]+C$$
$$\int \sqrt{a^{2}+x^{2}}dx=\frac{1}{2}\left [ x\sqrt{a^{2}+x^{2}}+a^{2}log(x+\sqrt{x^{2}+a^{2}}) \right ]+C$$
$$\int \sqrt{x^{2}-a^{2}}dx=\frac{1}{2}\left [ x\sqrt{x^{2}-a^{2}}-a^{2}log(x+\sqrt{x^{2}-a^{2}}) \right ]+C$$

'C' is the Integration constant.

### How to choose the first and second functions in Integration by parts

you can choose the first and second functions through a simple trick. Remember a word 'ILATE'

I → Inverse function (e.g.- sin-1 x, cos-1 x)

L → Logarithmic function (e.g.- log x)

A → Algebraic function (e.g.- x3, x4)

T → Trigonometric function (e.g.- sin x, cos x)

E → Exponatial function (e.g.- e2)

e.g.- $$\int x^2 \space logx \space dx$$

x2 → A → f2(x)

log x → L → f1(x)

$$=\int logx \space x^2 \space dx$$

$$=log x \int x^2 dx - \left [ \int \left ( \frac {d}{dx} log x \int x^2 \right ) dx \right ]$$

$$=logx \int x^2 dx - \left [ \int \frac {1}{x} \frac {x^3}{3} dx \right ]$$

$$=logx \frac {x^3}{3} - \left [ \int \frac {x^2}{3} dx \right ]$$

$$=\frac {x^3}{3} logx - \frac {x^3}{9}+c$$

## Definite Integration

### Properties of Definite Integrals

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

👉$$\int_{a}^{b}f(x)\hspace{1mm}dx=lim_{h\to 0}h\left [ f(a)+f(a+h)+f(a+2h)+.........+f\left\{a+(n-1)h \right\} \right ]$$

👉$$\int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx, when\hspace{1mm}a > b$$

👉$$\int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx, when\hspace{1mm}a < c < b$$

👉$$\int_{0}^{a}f(x)dx = \int_{0}^{a}f(a-x)dx+C$$

👉$$\int_{-a}^{a}f(x)dx =0, When\hspace {1mm}function \hspace {1mm}f(x), x\hspace {1mm} is\hspace {1mm} an\hspace {1mm} odd\hspace {1mm} function$$
$$\hspace {1.5cm}= 2\int_{0}^{a}f(x)dx, when\hspace {1mm} f(x), x\hspace {1mm} is\hspace {1mm} an\hspace {1mm} even\hspace {1mm} function$$
👉$$\int_{0}^{2a}f(x)dx =2\int_{0}^{a}f(x)dx, When\hspace {1mm} f(2a-x)=f(x)$$
$$\hspace {1.5cm} = 0, when f(2a-x) = -f(x)$$

👉 If $$\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$$ then,

⋆ $$\int_{0}^{b} f(x) dx = \int_{0}^{b} f(b-x) dx$$

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

👉 If f( a + b - x) = f(x) then,

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

'C' is the Integration constant.

### Definite integration Formula

$$\int_{0}^{\frac {π}{2}} log(sinx) dx = - \frac {π}{2} log 2$$

$$\int_{0}^{\frac {π}{2}} log(cosx) dx = - \frac {π}{2} log 2$$

$$\int_{0}^{\frac {π}{2}} log(secx) dx = \frac {π}{2} log 2$$

$$\int_{0}^{\frac {π}{2}} log(cosecx) dx = \frac {π}{2} log 2$$

$$\int_{0}^{\frac {π}{2}} log(tanx) dx = 0$$

$$\int_{0}^{\frac {π}{2}} log(cotx) dx = 0$$

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

$$\int_{a}^{b} \frac {1}{a^2cos^2x+b^2sin^2x}dx = \frac {π}{2ab}$$

### Leibnitz Theorem

If f (x) = $$\int_{g(x)}^{h(x)} f (t) dt$$ then,

$$\frac {d}{dx} f (x) = f \left ( h (x) \right ) \frac {d}{dx} h (x) - f \left ( g(x) \right ) \frac {d}{dx} g (x)$$

Don't forget to share with friends and support🙏