Integration

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Integration of the composite function

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

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

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

'C' is the Integration constant.

Integration formula of the Trigonometric function

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

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

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

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

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

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

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

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

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

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

'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

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

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

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

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

'C' is the Integration constant.

Integration of two functions or Integration by parts

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

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

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

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

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

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

'C' is the Integration constant.

Properties of Definite Integrals

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

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

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

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

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

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

'C' is the Integration constant.

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