What is the integral of x/(\sqrt[3]{x-2)} ?

The integral of x/(\sqrt[3]{x-2)} is 3/5 (x-2)^{5/3}+3(x-2)^{2/3}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x/(\sqrt[3]{x-2)}

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

\frac{3}{5} (x - 2)^{\frac{5}{3}} + 3 (x - 2)^{\frac{2}{3}} + C

Solution steps

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

Apply u-substitution

: \int \frac{u + 2}{3 u} d u

= \int \frac{u + 2}{3 u} d u

Expand

\frac{u + 2}{3 u} : u^{\frac{2}{3}} + \frac{2}{3 u}

= \int u^{\frac{2}{3}} + \frac{2}{3 u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{\frac{2}{3}} d u + \int \frac{2}{3 u} d u

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

\int \frac{2}{3 u} d u = 3 u^{\frac{2}{3}}

= \frac{3}{5} u^{\frac{5}{3}} + 3 u^{\frac{2}{3}}

Substitute back

u = x - 2

= \frac{3}{5} (x - 2)^{\frac{5}{3}} + 3 (x - 2)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} (x - 2)^{\frac{5}{3}} + 3 (x - 2)^{\frac{2}{3}} + C

x \to \frac{3 π}{2} lim (cos (x))

\int \frac{1}{9 - ( 3 x + 5 ) ^{2}} d x

y^{^{'''}} - 4 y^{^{''}} + y^{^{'}} + 6 y = 36

x \to - 1 lim (x^{2} + x)

\int_{2}^{5} \frac{1}{x - 2} d x

What is the integral of x/(\sqrt[3]{x-2)} ?
The integral of x/(\sqrt[3]{x-2)} is 3/5 (x-2)^{5/3}+3(x-2)^{2/3}+C