What is the integral of (8x^7)/(x^8-1) ?

The integral of (8x^7)/(x^8-1) is ln|x^8-1|+C

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integral of (8x^7)/(x^8-1)

\int \frac{8 x ^{7}}{x ^{8} - 1} d x

ln x^{8} - 1 + C

Solution steps

\int \frac{8 x ^{7}}{x ^{8} - 1} 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 ^{7}}{x ^{8} - 1} d x

Apply u-substitution

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

= 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 = x^{8} - 1

= 8 \cdot \frac{1}{8} ln x^{8} - 1

Simplify

8 \cdot \frac{1}{8} ln x^{8} - 1 : ln x^{8} - 1

= ln x^{8} - 1

Add a constant to the solution

= ln x^{8} - 1 + C

f (t) = e^{7 t \cdot s i n (2 t)}

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What is the integral of (8x^7)/(x^8-1) ?
The integral of (8x^7)/(x^8-1) is ln|x^8-1|+C