Find out the answer to "Evaluate the following problem. ∫x3(a2+x2)12dx"

waigaK

Answered question

2021-08-12

Evaluate the following problem.

\int x^{3} {(a^{2} + x^{2})}^{\frac{1}{2}} d x

Jayden-James Duffy

Skilled2021-08-13Added 91 answers

Our Aim is to evaluate the integral given below:

\int x^{3} (\sqrt{a^{2} + x^{2}}) d x - (i)

Considering the integral given below:

\int x^{3} (\sqrt{a^{2} + x^{2}}) d x - (i)

For the integrand

x^{2} \sqrt{a^{2} + x^{2}}

, substitute

u = x^{2}

and

d u = 2 x d x

= \frac{1}{2} u \sqrt{a^{2} + u} d u

For, the integrand

u \sqrt{a^{2} + u}

, substitute

s = a^{2} + u

and ds=du

\frac{1}{2} \int \sqrt{s} (s - a^{2}) d s

Expanding the integral

\sqrt{s} (s - a^{2})

given

s^{\frac{3}{2}} - a^{2} \sqrt{s}

= \frac{1}{2} \int (s^{\frac{3}{2}} - a^{2} \sqrt{s}) d s

Integrate the sum term by term and taking out constants:

= \frac{1}{2} \int s^{\frac{3}{2}} d s - \frac{a^{2}}{2} \int \sqrt{s} d s

The integral of

s^{\frac{3}{2}}

\frac{2 s^{\frac{5}{2}}}{5}

= \frac{s^{\frac{5}{2}}}{5} - \frac{a^{2}}{2} \int \sqrt{s} d s

The integral of

\sqrt{s}

\frac{2 s^{\frac{3}{2}}}{3}

= \frac{s^{\frac{5}{2}}}{5} - \frac{1}{3} a^{2} s^{\frac{3}{2}} + C

Substitute back for

s = a^{2} + u

= \frac{1}{5} {(a^{2} + u^{2})}^{\frac{5}{2}} - \frac{1}{3} a^{2} {(a^{2} + u)}^{\frac{3}{2}} + C

Substitute back for

u = x^{2}

This is helpful 1 Flag

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?