What is the integral of (x-1/(x^2))(x^2+2/x)^2 ?

The integral of (x-1/(x^2))(x^2+2/x)^2 is (x^6)/6+x^3+4/(3x^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-1/(x^2))(x^2+2/x)^2

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

\frac{x ^{6}}{6} + x^{3} + \frac{4}{3 x ^{3}} + C

Solution steps

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

Expand

(x - \frac{1}{x ^{2}}) (x^{2} + \frac{2}{x})^{2} : x^{5} + 3 x^{2} - \frac{4}{x ^{4}}

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

Apply the Sum Rule:

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

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

\int x^{5} d x = \frac{x ^{6}}{6}

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

\int \frac{4}{x ^{4}} d x = - \frac{4}{3 x ^{3}}

= \frac{x ^{6}}{6} + x^{3} - (- \frac{4}{3 x ^{3}})

Simplify

= \frac{x ^{6}}{6} + x^{3} + \frac{4}{3 x ^{3}}

Add a constant to the solution

= \frac{x ^{6}}{6} + x^{3} + \frac{4}{3 x ^{3}} + C

\int \frac{e ^{3 x}}{e ^{3 x} - 6} d x

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

\int \frac{sin ( x )}{a + b cos ( x )} d x

\int 4 x^{2} - 5 d x

\int \frac{2 x ^{2}}{7 - 6 x ^{3}} d x

What is the integral of (x-1/(x^2))(x^2+2/x)^2 ?
The integral of (x-1/(x^2))(x^2+2/x)^2 is (x^6)/6+x^3+4/(3x^3)+C