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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

\int \frac{2 x ^{2}}{1 + sin ( 8 ) x ^{3}} d x

\frac{2}{3 sin ( 8 )} ln 1 + sin (8) x^{3} + C

Solution steps

\int \frac{2 x ^{2}}{1 + sin ( 8 ) x ^{3}} d x

Take the constant out:

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

= 2 \cdot \int \frac{x ^{2}}{1 + sin ( 8 ) x ^{3}} d x

Apply u-substitution

= 2 \cdot \int \frac{1}{3 sin ( 8 ) u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{3 sin ( 8 )} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{3 sin ( 8 )} ln ∣ u ∣

Substitute back

u = 1 + sin (8) x^{3}

= 2 \cdot \frac{1}{3 sin ( 8 )} ln 1 + sin (8) x^{3}

Simplify

2 \cdot \frac{1}{3 sin ( 8 )} ln 1 + sin (8) x^{3} : \frac{2}{3 sin ( 8 )} ln 1 + sin (8) x^{3}

= \frac{2}{3 sin ( 8 )} ln 1 + sin (8) x^{3}

Add a constant to the solution

= \frac{2}{3 sin ( 8 )} ln 1 + sin (8) x^{3} + C

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