What is the integral of (x^2)/(sqrt((x^2+1)^5)) ?

The integral of (x^2)/(sqrt((x^2+1)^5)) is (x^3)/(3(1+x^2)^{3/2)}+C

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integral of (x^2)/(sqrt((x^2+1)^5))

\int \frac{x ^{2}}{( x ^{2} + 1 ) ^{5}} d x

\frac{x ^{3}}{3 ( 1 + x ^{2} ) ^{\frac{3}{2}}} + C

Solution steps

\int \frac{x ^{2}}{( x ^{2} + 1 ) ^{5}} d x

(x^{2} + 1)^{5} = (x^{2} + 1)^{\frac{5}{2}},

assuming

(x^{2} + 1) \geq 0

= \int \frac{x ^{2}}{( x ^{2} + 1 ) ^{\frac{5}{2}}} d x

Apply Trigonometric Substitution

: \int cos^{3} (u) tan^{2} (u) d u

= \int cos^{3} (u) tan^{2} (u) d u

Rewrite using trig identities

= \int sin^{2} (u) cos (u) d u

Apply u-substitution

: \int v^{2} d v

= \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{v ^{3}}{3}

Substitute back

= \frac{sin ^{3} ( arctan ( x ) )}{3}

Simplify

\frac{sin ^{3} ( arctan ( x ) )}{3} : \frac{x ^{3}}{3 ( 1 + x ^{2} ) ^{\frac{3}{2}}}

= \frac{x ^{3}}{3 ( 1 + x ^{2} ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{x ^{3}}{3 ( 1 + x ^{2} ) ^{\frac{3}{2}}} + C

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\int \frac{x ^{2} + 57}{x ^{2} + 64} d x

What is the integral of (x^2)/(sqrt((x^2+1)^5)) ?
The integral of (x^2)/(sqrt((x^2+1)^5)) is (x^3)/(3(1+x^2)^{3/2)}+C