What is the integral of x^{-5}(x+1) ?

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

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integral of x^{-5}(x+1)

\int x^{- 5} (x + 1) d x

- \frac{1}{3 x ^{3}} - \frac{1}{4 x ^{4}} + C

Solution steps

\int x^{- 5} (x + 1) d x

Simplify

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

= \int \frac{x + 1}{x ^{5}} d x

Expand

\frac{x + 1}{x ^{5}} : \frac{1}{x ^{4}} + \frac{1}{x ^{5}}

= \int \frac{1}{x ^{4}} + \frac{1}{x ^{5}} 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 ^{4}} d x + \int \frac{1}{x ^{5}} d x

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

\int \frac{1}{x ^{5}} d x = - \frac{1}{4 x ^{4}}

= - \frac{1}{3 x ^{3}} - \frac{1}{4 x ^{4}}

Add a constant to the solution

= - \frac{1}{3 x ^{3}} - \frac{1}{4 x ^{4}} + C

\frac{\partial}{\partial y} (x^{2} + y^{2} - x y)

x e^{x - 1}

\int e^{7 x} d x

\frac{d}{d x} (x e)

f (x) = 13 x

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