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

The integral of 1/((1+2x^2)^{5/2)} is (x(4x^2+3))/(3(2x^2+1)^{3/2)}+C

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integral of 1/((1+2x^2)^{5/2)}

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

\frac{x ( 4 x ^{2} + 3 )}{3 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}} + C

Solution steps

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

Apply Trigonometric Substitution

: \int \frac{1}{2 ^{\frac{1}{2}} sec ^{3} ( u )} d u

= \int \frac{1}{2 ^{\frac{1}{2}} sec ^{3} ( u )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2 ^{\frac{1}{2}}} \cdot \int \frac{1}{sec ^{3} ( u )} d u

Rewrite using trig identities

= \frac{1}{2 ^{\frac{1}{2}}} \cdot \int (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

: \int 1 - v^{2} d v

= \frac{1}{2 ^{\frac{1}{2}}} \cdot \int 1 - v^{2} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2 ^{\frac{1}{2}}} (\int 1 d v - \int v^{2} d v)

\int 1 d v = v

\int v^{2} d v = \frac{v ^{3}}{3}

= \frac{1}{2 ^{\frac{1}{2}}} (v - \frac{v ^{3}}{3})

Substitute back

= \frac{1}{2 ^{\frac{1}{2}}} sin (arctan (2^{\frac{1}{2}} x)) - \frac{sin ^{3} ( arctan ( 2 ^{\frac{1}{2}} x ) )}{3}

Simplify

\frac{1}{2 ^{\frac{1}{2}}} sin (arctan (2^{\frac{1}{2}} x)) - \frac{sin ^{3} ( arctan ( 2 ^{\frac{1}{2}} x ) )}{3} : \frac{x ( 4 x ^{2} + 3 )}{3 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}}

= \frac{x ( 4 x ^{2} + 3 )}{3 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{x ( 4 x ^{2} + 3 )}{3 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}}} + C

\int \frac{f ( ln ( x ) )}{x} d x

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What is the integral of 1/((1+2x^2)^{5/2)} ?
The integral of 1/((1+2x^2)^{5/2)} is (x(4x^2+3))/(3(2x^2+1)^{3/2)}+C