What is the integral of (32x^3)/(2x^4+5) ?

The integral of (32x^3)/(2x^4+5) is 4ln|2x^4+5|+C

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integral of (32x^3)/(2x^4+5)

\int \frac{32 x ^{3}}{2 x ^{4} + 5} d x

4 ln 2 x^{4} + 5 + C

Solution steps

\int \frac{32 x ^{3}}{2 x ^{4} + 5} d x

Take the constant out:

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

= 32 \cdot \int \frac{x ^{3}}{2 x ^{4} + 5} d x

Apply u-substitution

: \int \frac{1}{8 u} d u

= 32 \cdot \int \frac{1}{8 u} d u

Take the constant out:

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

= 32 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 32 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 2 x^{4} + 5

= 32 \cdot \frac{1}{8} ln 2 x^{4} + 5

Simplify

32 \cdot \frac{1}{8} ln 2 x^{4} + 5 : 4 ln 2 x^{4} + 5

= 4 ln 2 x^{4} + 5

Add a constant to the solution

= 4 ln 2 x^{4} + 5 + C

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What is the integral of (32x^3)/(2x^4+5) ?
The integral of (32x^3)/(2x^4+5) is 4ln|2x^4+5|+C