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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int \frac{x ^{3} - 2 x}{x} d x

\int 3 x^{3} e^{2} x d x

\int 54 (1 + x^{2}) (7 + 18 x + 6 x^{3})^{16} d x

\int \frac{5 e ^{4 u}}{e ^{4 u} + 3} d u

\int \frac{3 z}{3 - 7 z} d z

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