What is the integral of x/((x^2+2)sqrt(3x^2+5)) ?

The integral of x/((x^2+2)sqrt(3x^2+5)) is arctan(sqrt(3x^2+5))+C

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integral of x/((x^2+2)sqrt(3x^2+5))

\int \frac{x}{( x ^{2} + 2 ) 3 x ^{2} + 5} d x

arctan (3 x^{2} + 5) + C

Solution steps

\int \frac{x}{( x ^{2} + 2 ) 3 x ^{2} + 5} d x

Apply u-substitution

: \int \frac{1}{2 u ( u + 1 )} d u

= \int \frac{1}{2 u ( u + 1 )} d u

Take the constant out:

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

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

Apply u-substitution

: \int \frac{2}{v ^{2} + 1} d v

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

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} \cdot 2 arctan (v)

Substitute back

= \frac{1}{2} \cdot 2 arctan (3 x^{2} + 5)

Simplify

\frac{1}{2} \cdot 2 arctan (3 x^{2} + 5) : arctan (3 x^{2} + 5)

= arctan (3 x^{2} + 5)

Add a constant to the solution

= arctan (3 x^{2} + 5) + C

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