What is the integral of (8x^3)/(1+2x^4) ?

The integral of (8x^3)/(1+2x^4) is ln|1+2x^4|+C

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integral of (8x^3)/(1+2x^4)

\int \frac{8 x ^{3}}{1 + 2 x ^{4}} d x

ln 1 + 2 x^{4} + C

Solution steps

\int \frac{8 x ^{3}}{1 + 2 x ^{4}} d x

Take the constant out:

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

= 8 \cdot \int \frac{x ^{3}}{1 + 2 x ^{4}} d x

Apply u-substitution

= 8 \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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 \cdot \frac{1}{8} ln ∣ u ∣

Substitute back

u = 1 + 2 x^{4}

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

Simplify

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

= ln 1 + 2 x^{4}

Add a constant to the solution

= ln 1 + 2 x^{4} + C

What is the integral of (8x^3)/(1+2x^4) ?
The integral of (8x^3)/(1+2x^4) is ln|1+2x^4|+C