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

The integral of \sqrt[3]{x}(ln(x))^2 is 3/4 x^{4/3}ln^2(x)-3/2 (3/4 x^{4/3}ln(x)-9/16 x^{4/3})+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of \sqrt[3]{x}(ln(x))^2

\int 3 x (ln (x))^{2} d x

\frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}}) + C

Solution steps

\int 3 x (ln (x))^{2} d x

Simplify

= \int 3 x ln^{2} (x) d x

Apply Integration By Parts

: \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \int \frac{3}{2} x^{\frac{1}{3}} ln (x) d x

= \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \int \frac{3}{2} x^{\frac{1}{3}} ln (x) d x

\int \frac{3}{2} x^{\frac{1}{3}} ln (x) d x = \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}})

= \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}})

Add a constant to the solution

= \frac{3}{4} x^{\frac{4}{3}} ln^{2} (x) - \frac{3}{2} (\frac{3}{4} x^{\frac{4}{3}} ln (x) - \frac{9}{16} x^{\frac{4}{3}}) + C

\int 6 x 6 x^{2} + 19 d x

\int (1 + t^{3})^{4} d t

\int \frac{x}{5 + 2 x ^{2}} d x

\int \frac{4 x ^{3} - 4 x}{x ^{4} - 2 x ^{2} + 5} d x

\int e^{s e c (x)} sec (x) tan (x) d x

What is the integral of \sqrt[3]{x}(ln(x))^2 ?
The integral of \sqrt[3]{x}(ln(x))^2 is 3/4 x^{4/3}ln^2(x)-3/2 (3/4 x^{4/3}ln(x)-9/16 x^{4/3})+C