What is the integral of 4/(sqrt((x^2+2)^3)) ?

The integral of 4/(sqrt((x^2+2)^3)) is (2x)/(sqrt(2+x^2))+C

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integral of 4/(sqrt((x^2+2)^3))

\int \frac{4}{( x ^{2} + 2 ) ^{3}} d x

\frac{2 x}{2 + x ^{2}} + C

Solution steps

\int \frac{4}{( x ^{2} + 2 ) ^{3}} d x

Take the constant out:

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

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

Apply Integral Substitution

= 4 \cdot \int \frac{1}{2 ( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

: \int cos (v) d v

= 4 \cdot \frac{1}{2} \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 4 \cdot \frac{1}{2} sin (v)

Substitute back

= 4 \cdot \frac{1}{2} sin (arctan (\frac{x}{2}))

Simplify

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

= \frac{2 x}{2 + x ^{2}}

Add a constant to the solution

= \frac{2 x}{2 + x ^{2}} + C

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What is the integral of 4/(sqrt((x^2+2)^3)) ?
The integral of 4/(sqrt((x^2+2)^3)) is (2x)/(sqrt(2+x^2))+C