Integrate the following with respect to $x$ 

$x \sin 3x$

1. Newton-Leibnitz integral:

A function $F\left(x\right)$ is called an antiderivative of a function $f\left(x\right)$ on an interval $I$ if $F^{\text{'}}\left(x\right)=f\left(x\right)$ for every value of $x$ in $I$.

2. Fundamental Integration Formula

(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, $\int f\left(ax+b\right)dx=\frac{F\left(ax+b\right)}{a}+C$

3. 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

4. Integration by the Method of Substitution:

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

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

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

5. Integration by the Method of Partial Fractions:

If the integrand is in the form of an algebraic fraction and the integral cannot be evaluated by simple methods, then the fraction needs to be expressed in partial fractions before integration takes place.

6. Integrals of the form $\mathbf{\int }{\mathit{e}}^{\mathit{a}\mathit{x}}\mathbf{sin}\mathit{b}\mathit{x}\mathbf{ }\mathit{d}\mathit{x}$ and $\mathbf{\int }{\mathit{e}}^{\mathit{a}\mathit{x}}\mathbf{cosb}\mathit{x}\mathbf{ }\mathit{d}\mathit{x}$:

(i) $\int e^{ax}\sin bx dx=\frac{e^{ax}}{a^2+b^2}[a\sin bx-b\cos bx]+C$

(ii) $\int e^{ax}\cos bx dx=\frac{e^{ax}}{a^2+b^2}[a\cos bx+b\sin bx]+C$

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

8. 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.

9. Integrals of the form $\mathbf{\int }\frac{\mathit{p}\mathbf{c}\mathbf{o}\mathbf{s}\mathit{x}\mathbf{+}\mathit{q}\mathbf{s}\mathbf{i}\mathbf{n}\mathit{x}}{\mathit{a}\mathbf{c}\mathbf{o}\mathbf{s}\mathit{x}\mathbf{+}\mathit{b}\mathbf{s}\mathbf{i}\mathbf{n}\mathit{x}}\mathit{d}\mathit{x}\mathbf{:}$

Put $p \cos x+q \sin x=A\left(a \cos x+b \sin x\right)+B\frac{d}{dx}\left(a \cos x+b \sin x\right)$, where $A$ and $B$ are constants to be found by solving the equations obtained by equating the corresponding coefficients of $\sin x$ and $\cos x$ on both sides.

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