What is the integral of x^{-3}(x+1) ?

The integral of x^{-3}(x+1) is -1/x-1/(2x^2)+C

integral of x^{-3}(x+1)

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

- \frac{1}{x} - \frac{1}{2 x ^{2}} + C

Solution steps

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

Simplify

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

= \int \frac{x + 1}{x ^{3}} d x

Expand

\frac{x + 1}{x ^{3}} : \frac{1}{x ^{2}} + \frac{1}{x ^{3}}

= \int \frac{1}{x ^{2}} + \frac{1}{x ^{3}} 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{1}{x ^{2}} d x + \int \frac{1}{x ^{3}} d x

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

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

= - \frac{1}{x} - \frac{1}{2 x ^{2}}

Add a constant to the solution

= - \frac{1}{x} - \frac{1}{2 x ^{2}} + C

\int_{0}^{1} 8^{2 x} d x

d er i v a t i v e 5 x sin (x^{2})

\int \frac{e ^{x}}{x} d x

d er i v a t i v e f (x) = 8 x e^{x} - 8 e^{x}

\frac{d}{d x} (x 4^{x})

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