# How do you integrate int ( 6x^3+3x+sqrt7 ) sin( 5x ) dx?

Oct 23, 2015

$I = \left(- \frac{6 {x}^{3}}{5} - \frac{39 x}{125} - \frac{\sqrt{7}}{5}\right) \cos 5 x + \left(\frac{18 {x}^{2}}{25} - \frac{21}{625}\right) \sin 5 x + C$

#### Explanation:

Let's solve (using Integration By Parts):

${I}_{n} = \int {x}^{n} \sin a x \mathrm{dx}$

$u = {x}^{n} \implies \mathrm{du} = n {x}^{n - 1} \mathrm{dx}$

$\mathrm{dv} = \sin a x \mathrm{dx} \implies v = \int \sin a x \mathrm{dx} = - \frac{1}{a} \cos a x$

${I}_{n} = - {x}^{n} / a \cos a x + \frac{n}{a} \int {x}^{n - 1} \cos a x \mathrm{dx}$

$u = {x}^{n - 1} \implies \mathrm{du} = \left(n - 1\right) {x}^{n - 2} \mathrm{dx}$

$\mathrm{dv} = \cos a x \mathrm{dx} \implies v = \int \cos a x \mathrm{dx} = \frac{1}{a} \sin a x$

${I}_{n} = - {x}^{n} / a \cos a x + \frac{n}{a} \left[{x}^{n - 1} / a \sin a x - \frac{n - 1}{a} \int {x}^{n - 2} \sin a x \mathrm{dx}\right]$

${I}_{n} = - {x}^{n} / a \cos a x + \frac{n {x}^{n - 1}}{a} ^ 2 \sin a x - \frac{n \left(n - 1\right)}{a} ^ 2 {I}_{n - 2}$

This is recurrent formula, we need to find ${I}_{1}$ and ${I}_{2}$ in general case. For our task, we need ${I}_{1}$:

${I}_{1} = \int x \sin a x \mathrm{dx}$

$u = x \implies \mathrm{du} = \mathrm{dx}$

$\mathrm{dv} = \sin a x \mathrm{dx} \implies v = \int \sin a x \mathrm{dx} = - \frac{1}{a} \cos a x \mathrm{dx}$

${I}_{1} = - \frac{x}{a} \cos a x + \frac{1}{a} \int \cos a x \mathrm{dx} = - \frac{x}{a} \cos a x + \frac{1}{a} ^ 2 \sin a x$

$a = 5 \implies {I}_{1} = - \frac{x}{5} \cos 5 x + \frac{1}{25} \sin 5 x$

$n = 3 \implies {I}_{3} = - {x}^{3} / 5 \cos 5 x + \frac{3 {x}^{2}}{25} \sin 5 x - \frac{6}{25} {I}_{1}$

${I}_{3} = - {x}^{3} / 5 \cos 5 x + \frac{3 {x}^{2}}{25} \sin 5 x - \frac{6}{25} \left(- \frac{x}{5} \cos 5 x + \frac{1}{25} \sin 5 x\right)$

${I}_{3} = - {x}^{3} / 5 \cos 5 x + \frac{3 {x}^{2}}{25} \sin 5 x + \frac{6 x}{125} \cos 5 x - \frac{6}{625} \sin 5 x$

$I = \int \left(6 {x}^{3} + 3 x + \sqrt{7}\right) \sin 5 x \mathrm{dx}$

$I = 6 \int {x}^{3} \sin 5 x \mathrm{dx} + 3 \int x \sin 5 x \mathrm{dx} + \sqrt{7} \int \sin 5 x \mathrm{dx}$

$I = 6 {I}_{3} + 3 {I}_{1} - \frac{\sqrt{7}}{5} \cos 5 x + C$

$I = 6 \left(- {x}^{3} / 5 \cos 5 x + \frac{3 {x}^{2}}{25} \sin 5 x + \frac{6 x}{125} \cos 5 x - \frac{6}{625} \sin 5 x\right) + 3 \left(- \frac{x}{5} \cos 5 x + \frac{1}{25} \sin 5 x\right) - \frac{\sqrt{7}}{5} \cos 5 x + C$

$I = \left(- \frac{6 {x}^{3}}{5} + \frac{36 x}{125} - \frac{3 x}{5} - \frac{\sqrt{7}}{5}\right) \cos 5 x + \left(\frac{18 {x}^{2}}{25} - \frac{36}{625} + \frac{3}{25}\right) \sin 5 x + C$

$I = \left(- \frac{6 {x}^{3}}{5} - \frac{39 x}{125} - \frac{\sqrt{7}}{5}\right) \cos 5 x + \left(\frac{18 {x}^{2}}{25} - \frac{21}{625}\right) \sin 5 x + C$