Find the integral of the function ${\sin }^3(2x+1)$

(i) ∫xndx=xn+1n+1+C, ∀n≠-1

(ii) $\int x^{-1}dx=\int \frac{1}{x}dx=\mathrm{l}\mathrm{o}\mathrm{g}\left|x\right|+C$

(iii) ∫exdx=ex+C

(iv) ∫axdx=axloga+C

(v) $\int \mathrm{l}\mathrm{o}\mathrm{g}x dx=x\mathrm{l}\mathrm{o}\mathrm{g}x-x+C$

(vi) ∫1x2dx=-1x+C

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

(viii) $\int \mathrm{s}\mathrm{i}\mathrm{n}x dx=-\mathrm{c}\mathrm{o}\mathrm{s}x+C$

(ix) $\int \cos x dx=\mathrm{s}\mathrm{i}\mathrm{n}x+C$

(x) $\int \mathrm{t}\mathrm{a}\mathrm{n}x dx=\mathrm{l}\mathrm{o}\mathrm{g}\left|\mathrm{s}\mathrm{e}\mathrm{c}x\right|+C$

(xi) $\int \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}x dx=\mathrm{l}\mathrm{o}\mathrm{g}\left|\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}x-\mathrm{c}\mathrm{o}\mathrm{t}x\right|+C$

(xii) $\int \mathrm{s}\mathrm{e}\mathrm{c}x dx=\mathrm{l}\mathrm{o}\mathrm{g}\left|\mathrm{s}\mathrm{e}\mathrm{c}x+\mathrm{t}\mathrm{a}\mathrm{n}x\right|+C$

(xiii) $\int \mathrm{c}\mathrm{o}\mathrm{t}x dx=\mathrm{l}\mathrm{o}\mathrm{g}\left|\mathrm{s}\mathrm{i}\mathrm{n}x\right|+C$

(xiv) ∫11-x2dx=sin-1x+C

(xv) ∫11+x2dx=tan-1x+C

(xvi) $\int \frac{1}{x\sqrt{x^2-1}}dx=\mathrm{s}\mathrm{e}{\mathrm{c}}^{-1}x+C$

(xvii) $\int \mathrm{s}\mathrm{e}{\mathrm{c}}^{2}x dx=\mathrm{t}\mathrm{a}\mathrm{n}x+C$

(xviii) $\int \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}x \mathrm{c}\mathrm{o}\mathrm{t}x dx=-\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}x+C$

(ix) $\int \mathrm{s}\mathrm{e}\mathrm{c}x\mathrm{t}\mathrm{a}\mathrm{n}x dx=\mathrm{s}\mathrm{e}\mathrm{c}x+C$

(xx) $\int \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}{\mathrm{c}}^{2}x dx=-\mathrm{c}\mathrm{o}\mathrm{t}x+C$

Note: If ∫f(x)dx=F(x)+C, then, $\int f\left(ax+b\right)dx=\frac{F(ax+b)}{a}+C$

2. Some Standard Results of Integration:

(i) ∫1x2+a2dx=log⁡x+x2+a2+C

(ii) ∫1x2-a2dx=log⁡x+x2-a2+C

(iii) ∫1x2-a2dx=12alog⁡x-ax+a+C

(iv) ∫1a2-x2dx=12alog⁡a+xa-x+C

(v) ∫a2-x2dx=x2a2-x2+a22sin-1xa+C

(vi) ∫a2+x2dx=x2a2+x2+a22log⁡x+a2+x2+C

(vii) ∫x2-a2dx=x2x2-a2-a22log|x+x2-a2|+C

3. Integration by the Method of Substitution:

To integrate, the integrals of the form:$\int {\left[f\right(x\left)\right]}^n{f}^{\text{'}}\left(x\right)dx$,

Put $f\left(x\right)=t$ $\Rightarrow f^{\text{'}}\left(x\right)dx=dt$

$\therefore \int {\left[f\left(x\right)\right]}^n{f}^{\text{'}}\left(x\right)dx=\int t^{n}dt=\frac{t^{n+1}}{n+1}+C=\frac{{\left[f\left(x\right)\right]}^{n+1}}{n+1}+C$ 

4. Integration by the Method of Partial Fractions:

An algebraic rational function of $x$ is a function of the form fxgx where $f\left(x\right)$ and $g\left(x\right)$ are both polynomial functions and g(x)≠0.

5. Integrals of the Form$\int \frac{\mathit{p}\mathit{x}+\mathit{q}}{\mathit{a}{\mathit{x}}^2+\mathit{b}\mathit{x}+\mathit{c}}\mathit{d}\mathit{x}$:

Put $px+q=l\frac{d}{dx}\left(a{x}^2+bx+c\right)+m=l(2ax+b)+m$, where $l$ and $m$ are determined by comparing coefficients on both sides.

6. Integrals of the Form ∫dxacosx+bsinx+c:

Express $\sin x$ and $\cos x$ in terms of $\tan \frac{x}{2}$. And then substitute tanx2=t.

7. Integrals of the form $\int \frac{p\cos x+q\sin x}{a\cos x+b\sin x}dx:$

Put $p\hspace{0.17em}\cos \hspace{0.17em}x+q\hspace{0.17em}\sin \hspace{0.17em}x=l\left(a\hspace{0.17em}\cos x+b\hspace{0.17em}\sin x\right)+m\frac{d}{dx}\left(a\hspace{0.17em}\cos x+b\hspace{0.17em}\sin x\right)$ 

where $l$ and $m$ are constants to be found by solving the equations obtained by equating the corresponding coefficients of $\sin x$ and $\cos x$ on both sides.

8. Integration by Parts:

(i) $\int f_1\left(x\right)·f_2\left(x\right)dx=f_1\left(x\right)\int f_2\left(x\right)dx-\int \left[\frac{d}{dx}{f}_1\left(x\right)·\int f_2\left(x\right)dx\right]dx$, where $f_1\left(x\right)$ is considered as the $I$ function and $f_2\left(x\right)$ as the $II$ function.

We generally follow the rule ‘ILATE’ to determine which of the two functions is to be $I$ and which of the two functions is to be the $II$.

(ii) $\int e^x\left[f\left(x\right)+f^{\text{'}}\left(x\right)\right] dx=e^{x}f\left(x\right)+C$

9. First fundamental Theorem of Integral Calculus:

Let $f$ be a continuous function of $x$ for $a\le x\le b$ and $A\left(x\right)={\int }_a^{x}f\left(x\right)dx$, then $A^{\text{'}}\left(x\right)=f\left(x\right)$ for all $x$ in $\left[a,b\right]$ and $A\left(a\right)=0$.

10. Second fundamental Theorem of Integral Calculus:

Let $\varphi \left(x\right)$ be the primitive or anti-derivative of a function $f\left(x\right)$ defined on $[a, b]$ i.e., ddx$\left(\varphi \right(x\left)\right)=f\left(x\right).$ Then, $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\varphi \left(b\right)-\varphi \left(a\right)$

11. Following are some fundamental properties of definite integrals that are very useful in evaluating integrals:

(i) $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f\left(t\right)dt$

(ii) $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=-\underset{b}{\overset{a}{\int }}f\left(x\right)dx$

(iii) $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{c}{\int }}f\left(x\right)dx+\underset{c}{\overset{b}{\int }}f\left(x\right)dx$

(iv) $\begin{array}{l}\underset{0}{\overset{a}{\int }}f\left(x\right)dx=\underset{0}{\overset{a}{\int }}f\end{array}\left(a-x\right)dx$

(v) $\underset{-a}{\overset{a}{\int }}f\left(x\right)dx=\left\{\begin{array}{l}2\underset{0}{\overset{a}{\int }}\mathrm{ }\mathrm{ }f\left(x\right)dx\mathrm{ } ,\mathrm{ }\mathrm{ }\mathrm{ }\text{ if } f\left(x\right) \text{ is an even function}\\ 0\mathrm{ }\mathrm{ } \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\text{ if } f\left(x\right) \text{is an odd function}\end{array}\right.$

(vi) $\underset{-a}{\overset{a}{\int }}\mathrm{ }f\left(x\right)dx=\underset{0}{\overset{a}{\int }}\mathrm{ }\left\{f\left(x\right)+f(-x)\right\}dx$

(vii) $\underset{0}{\overset{2a}{\int }}f\left(x\right)dx=\left\{\begin{array}{l}2\underset{0}{\overset{a}{\int }}\mathrm{ }f\left(x\right)dx,\mathrm{ }\mathrm{ }\mathrm{ } \text{if} f\left(2a-x\right)=f\left(x\right)\\ 0\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ } \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ },\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ } \text{if} f\left(2a-x\right)=-f\left(x\right)\end{array}\right.$

(viii) $\underset{a}{\overset{2a}{\int }}\mathrm{ }\mathrm{ }f\left(x\right)dx=\underset{0}{\overset{2a}{\int }}\mathrm{ }\left\{f\left(x\right)+f(2a-x)\right\}dx$

 (ix) $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)dx$

 12. Definite Integral as the limit of a sum:

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{ }h\left[f\left(a\right)+f(a+h)+f(a+2h)+\dots +f\left\{(a+(n-1)h\right\}\right]$

                             or

${\int }_a^{b}f\left(x\right)dx=\underset{h\to 0}{\lim }h\left[f\left(a+h\right)+f\left(a+2h\right)+f\left(a+3h\right)+\cdots ++f\left(a+nh\right)\right]$

where $h=\frac{b-a}{n}$.

13. Leibniz's integral rule:

$\frac{d}{dx}{\int }_{g\left(x\right)}^{h\left(x\right)}f\left(t\right)dt=f\left(h\left(x\right)\right)h^{\text{'}}\left(x\right)-f\left(g\left(x\right)\right)g^{\text{'}}\left(x\right)$