What is the integral of sqrt(x)((-3)/(x^3)-x^2+x/4) ?

The integral of sqrt(x)((-3)/(x^3)-x^2+x/4) is 2/(x^{3/2)}+1/10 x^{5/2}-2/7 x^{7/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(x)((-3)/(x^3)-x^2+x/4)

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

\frac{2}{x ^{\frac{3}{2}}} + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C

Solution steps

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

Expand

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

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

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

\int \frac{x ^{\frac{3}{2}}}{4} d x = \frac{1}{10} x^{\frac{5}{2}}

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= - (- \frac{2}{x ^{\frac{3}{2}}}) + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}}

Simplify

= \frac{2}{x ^{\frac{3}{2}}} + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}}

Add a constant to the solution

= \frac{2}{x ^{\frac{3}{2}}} + \frac{1}{10} x^{\frac{5}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C

\int \frac{sin ( x ) - cot ( x )}{sin ^{2} ( x )} d x

\int (3 x^{2} + 1) e^{x^{3} + x} d x

\int \frac{1}{x - 2 x} d x

\int \frac{2 x ^{3} + 8 x}{x ^{2} + 4} d x

What is the integral of sqrt(x)((-3)/(x^3)-x^2+x/4) ?
The integral of sqrt(x)((-3)/(x^3)-x^2+x/4) is 2/(x^{3/2)}+1/10 x^{5/2}-2/7 x^{7/2}+C