Antiderivative (Indefinite Integral)

Consider a continuous function $ƒ$ defined in an open interval $(a,b)$. Then a function $F$ is said to be an antiderivative of $ƒ$ on the interval, if the derivative of $F$ is equal to $ƒ$ on the interval i.e. \[\text{if }\frac{dF(x)}{dx}=ƒ(x),x\in(a,b)\]

As the derivative of a constant is zero, $F(x)+C$ is also an antiderivative of $ƒ$. Thus, when $\frac{dF(x)}{dx}=ƒ(x)$, we have \[\frac{d[F(x)+C]}{dx}=\frac{dF(x)}{dx}+\frac{dC}{dx}\] \[=ƒ(x)+0\] \[=ƒ(x)\]

Any two antiderivatives of a function also differ by a constant. Let $F$ and $G$ be the antiderivatives of a function $ƒ$. Then, \[\frac{d[F(x)-G(x)]}{dx}=\frac{dF(x)}{dx}-\frac{dG(x)}{dx}\] \[=ƒ(x)-ƒ(x)\] \[=0\] Hence, from this, it follows that there exists a constant $C$ such that \[F(x)-G(x)=C\]

Hence, all the above mentioned things establish the fact that if $F$ is an antiderivative of $ƒ$, $F(x)+C$ gives all the possible antiderivatives of $ƒ$, when $C$ runs through all real numbers.

The general form of all antiderivatives of $ƒ$, which we call indefinite integral of $ƒ$, is denoted by \[\int ƒdx \text{ or } \int ƒ(x)dx\] If $F$ is an antiderivative of $ƒ$, we have \[\int ƒ(x)dx=F(x)+C\]

One basic property of the indefinite integral is \[\int [c_1ƒ(x)+c_2g(x)]dx\]\[=c_1\int ƒ(x)dx+c_2\int g(x)dx+C\] where $ƒ$ and $g$ are continuous functions in an interval $(a,b)$ and $c_1$ and $c_2$ are some constants.

Some Integration Formulae

\[\int 1dx=x+C\] \[\int x^ndx=\frac{x^{n+1}}{n+1}+C\text{ (n≠-1)}\] \[\int\sin axdx=-\frac{\cos ax}{a}+C\] \[\int\cos axdx=\frac{\sin ax}{a}+C\] \[\int\sec^2axdx=\frac{\tan ax}{a}+C\] \[\int\operatorname{cosec}^2axdx=-\frac{\cot ax}{a}+C\] \[\int\sec ax\tan ax dx=\frac{\sec ax}{a}+C\] \[\int\operatorname{cosec}ax\cot axdx=-\frac{\operatorname{cosec}ax}{a}+C\] \[\int(ax+b)^ndx=\frac{(ax+b)^{n+1}}{(n+1)a}+C\] \[\int\frac{1}{x}dx=\log x+C\] \[\int e^{ax}dx=\frac{e^{ax}}{a}+C\] \[\int\frac{1}{ax+b}dx=\frac{\log(ax+b)}{a}+C\]


Find the indefinite integrals:

$\int 5x^3dx$

\[\int 5x^3dx\] \[=5\int x^3dx\] \[=\frac{5}{4}x^4+C\]


$\int (2x+1)(3x+2)dx$

\[\int (2x+1)(3x+2)dx\] \[=\int (6x^2+7x+2)dx\] \[=\int 6x^2dx+\int 7xdx+\int 2 dx\] \[=6\frac{x^3}{3}+7\frac{x^2}{2}+2x+C\] \[=2x^3+\frac{7}{2}x^2+2x+C\]


$\int\frac{ax^2+bx+c}{x^2}dx$

\[\int\frac{ax^2+bx+c}{x^2}dx\] \[=\left(a+\frac{b}{x}+\frac{c}{x^2}\right)dx\] \[=\int adx+\int\frac{b}{x}dx+\frac{c}{x^2}dx\] \[=a\int 1dx+b\int\frac{1}{x}dx+c\int\frac{1}{x^2}dx\] \[=ax+b\log x+c\frac{x^{-1}}{-1}+C\] \[=ax+b\log x-\frac{c}{x}+C\]


$\int(a-bx)^5dx$

\[\int(a-bx)^5dx\] \[=\frac{(a-bx)^6}{6×(-b)}+C\] \[=-\frac{1}{6b}(a-bx)^6+C\]


$\int\frac{x^2+5}{x+2}dx$

\[\int\frac{x^5+5}{x+2}dx\] \[=\int\frac{\overline{x^2-4}+9}{x+2}dx\] \[=\int\frac{(x+2)(x-2)+9}{x+2}dx\] \[=\int(x-2)dx+\int\frac{9}{x+2}dx\] \[=\int xdx-\int 2dx+9\frac{1}{x+2}dx\] \[=\frac{x^2}{2}-2x+9\log(x+2)+C\]


$\int 2x\sqrt{2x+3}dx$

\[\int 2x\sqrt{2x+3}dx\] \[=\int (\overline{2x+3}-3)\sqrt{2x+3}dx\] \[=\int(2x+3)^{\frac{3}{2}}dx-\int 3(2x+3)^{\frac{1}{2}}dx\] \[=\frac{(2x+3)^{\frac{5}{2}}}{\frac{5}{2}×2}-3\frac{(2x+3)^{\frac{3}{2}}}{\frac{3}{2}×2}+C\] \[=\frac{1}{5}(2x+3)^{\frac{5}{2}}-(2x+3)^{\frac{3}{2}}+C\]


$\int\frac{3x+2}{\sqrt{5x+3}}dx$

\[\int\frac{3x+2}{\sqrt{5x+3}}dx\] \[=3\int\frac{x+\frac{2}{3}}{\sqrt{5x+3}}dx\] \[=\frac{3}{5}\int\frac{5x+\frac{10}{3}}{\sqrt{5x+3}}dx\] \[=\frac{3}{5}\int\frac{\overline{5x+3}+\frac{1}{3}}{\sqrt{5x+3}}dx\] \[=\frac{3}{5}\int(5x+3)^{\frac{1}{2}}dx+\frac{3}{5}\int\frac{1}{3}(5x+3)^{-\frac{1}{2}}dx\] \[=\frac{3}{5}\frac{(5x+3)^{\frac{3}{2}}}{\frac{3}{2}×5}+\frac{1}{5}\frac{(5x+3)^{\frac{1}{2}}}{\frac{1}{2}×5}+C\] \[=\frac{2}{25}(5x+3)^{\frac{3}{2}}+\frac{2}{25}(5x+3)^{\frac{1}{2}}+C\]


$\int\frac{dx}{\sqrt{x+a}-\sqrt{x-b}}$

\[\int\frac{dx}{\sqrt{x+a}-\sqrt{x-b}}\] \[=\int\frac{\sqrt{x+a}+\sqrt{x-b}}{x+a-x+b}dx\] \[=\int\frac{\sqrt{x+a}+\sqrt{x-b}}{a+b}dx\]\[=\frac{1}{a+b}\left[\int(x+a)^{\frac{1}{2}}dx+\int(x-b)^{\frac{1}{2}}dx\right]\] \[=\frac{1}{a+b}\left[\frac{(x+a)^{\frac{3}{2}}}{\frac{3}{2}}+\frac{(x-b)^{\frac{3}{2}}}{\frac{3}{2}}\right]+C\] \[=\frac{2}{3(a+b)}\left[(x+a)^{\frac{3}{2}}+(x-b)^{\frac{3}{2}}\right]+C\]


$\int\cos^2bxdx$

\[\int\cos^2bxdx\] \[=\frac{1}{2}\int 2\cos^2bxdx\] \[=\frac{1}{2}\int(1+\cos 2bx)dx\] \[=\frac{1}{2}\int 1dx+\frac{1}{2}\int\cos 2bx dx\] \[=\frac{1}{2}x+\frac{1}{2}\frac{\sin 2bx}{2b}+C\] \[=\frac{x}{2}+\frac{1}{4b}\sin 2bx+C\]


$\int\frac{1}{\cos^2x\sin^2x}dx$

\[\int\frac{1}{\cos^2x\sin^2x}dx\] \[=\int\frac{\sin^2x+\cos^2x}{\cos^2x\sin^2x}dx\] \[=\int\sec^2xdx+\int\operatorname{cosec}^2xdx\] \[=\tan x-\cot x+C\]


$\int\sqrt{1+\cos nx}dx$

\[\int\sqrt{1+\cos nx}dx\] \[=\int\sqrt{2\cos^2\frac{nx}{2}}dx\] \[=\sqrt{2}\int\cos\frac{nx}{2}dx\] \[=\sqrt{2}\frac{\sin\frac{nx}{2}}{\frac{n}{2}}+C\] \[=\frac{2\sqrt{2}}{n}\sin\frac{nx}{2}+C\]


$\int\frac{dx}{1-\sin ax}$

\[\int\frac{dx}{1-\sin ax}\] \[=\int\frac{1+\sin ax}{\cos^2ax}dx\] \[=\int\sec^2axdx+\int\sec ax\tan ax dx\] \[=\frac{\tan ax}{a}+\frac{\sec ax}{a}+C\]


$\int\cos px\cos qxdx$

\[\int\cos px\cos qxdx\] \[=\frac{1}{2}\int 2\cos px\cos qxdx\] \[=\frac{1}{2}\int[\cos(px-qx)+\cos(px+qx)]dx\] \[=\frac{1}{2}\int[\cos(p-q)x+\cos(p+q)x]dx\] \[=-\frac{1}{2}\frac{\sin(p-q)x}{p-q}-\frac{1}{2}\frac{\sin(p+q)x}{p+q}+C\] \[=-\frac{\sin(p-q)x}{2(p-q)}-\frac{\sin(p+q)x}{2(p+q)}+C\]


$\int(e^{px}+e^{-px})^2dx$

\[\int(e^{px}+e^{-px})^2dx\] \[=\int(e^{2px}+2+e^{-2px})dx\] \[=\int e^{2px}dx+2\int 1dx+\int e^{-2px}dx\] \[=\frac{e^{2px}}{2p}+2x+\frac{e^{-2px}}{-2p}+C\] \[=\frac{1}{2p}e^{2px}+2x-\frac{1}{2p}e^{-2px}+C\]


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