Integrate by making substitution and by parts

[find_the_fun]

I don't understand the question:

"First make a substitution and then use integration by parts to evaluate the integral"

\(\displaystyle \int sin \sqrt{x} dx\)

What does it have in mind by "substitution"?

 

[Dethrone]

Hi find_the_fun, (Wave)

By "substitution", they are referring to a variable substitution i.e u-substitution. What substitution would get rid of the square root? (Wondering)

 

[find_the_fun]

Hi find_the_fun, (Wave)

By "substitution", they are referring to a variable substitution i.e u-substitution. What substitution would get rid of the square root? (Wondering)

So let \(\displaystyle u=\sqrt{x}\) then \(\displaystyle du=\frac{1}{2\sqrt{x}}dx\)
but none of that's in the original equation?

 

[Dethrone]

I would use the substitution $x=u^2$. This will eliminate the square root from the argument. (Nod)

I wouldn't square root both sides at this step, I would take the differential of both sides.
$dx =2u\cdot du$.

Can you proceed from here?

They are effectively the same, but taking the square root introduces more steps:

$$\sqrt{x}=u$$
$$\frac{1}{2\sqrt{x}}dx=du$$
$$dx=2\sqrt{x}\cdot du$$
Note that $\sqrt{x} = u$
$$dx=2u\cdot du$$

 

[Prove It]

I don't understand the question:

"First make a substitution and then use integration by parts to evaluate the integral"

\(\displaystyle \int sin \sqrt{x} dx\)

What does it have in mind by "substitution"?

$\displaystyle \begin{align*} \int{ \sin{ \left( \sqrt{x} \right) } \,\mathrm{d}x } &= \int{ \frac{2\,\sqrt{x}\,\sin{ \left( \sqrt{x} \right) } }{2\,\sqrt{x}}\,\mathrm{d}x } \end{align*}$

Now let $\displaystyle \begin{align*} u = \sqrt{x} \implies \mathrm{d}u = \frac{1}{2\,\sqrt{x}}\,\mathrm{d}x \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} \int{ \frac{2\,\sqrt{x}\,\sin{ \left( \sqrt{x} \right) }}{2\,\sqrt{x}}\,\mathrm{d}x } &= \int{ 2\,u\sin{(u)}\,\mathrm{d}u} \\ &= -2\,u\cos{(u)} - \int{ -2\cos{(u)}\,\mathrm{d}u } \\ &= -2\,u\cos{(u)} + 2\int{ \cos{(u)}\,\mathrm{d}u } \\ &= -2\,u\cos{(u)} + 2\sin{(u)} + C \\ &= -2\,\sqrt{x}\,\cos{ \left( \sqrt{x} \right) } + 2\sin{ \left( \sqrt{x} \right) } + C \end{align*}$