What is the integral of (3x^2+2x)/(x^6) ?

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

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integral of (3x^2+2x)/(x^6)

integral

\frac{3 x ^{2} + 2 x}{x ^{6}}

- \frac{1}{x ^{3}} - \frac{1}{2 x ^{4}} + C

Solution steps

\int \frac{3 x ^{2} + 2 x}{x ^{6}} d x

Simplify

\frac{3 x ^{2} + 2 x}{x ^{6}} : \frac{3 x + 2}{x ^{5}}

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

Expand

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

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

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

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

= - \frac{1}{x ^{3}} - \frac{1}{2 x ^{4}}

Add a constant to the solution

= - \frac{1}{x ^{3}} - \frac{1}{2 x ^{4}} + C

1 + 2 x

(1 + 2 cos (x))^{2}

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

(2 x^{2})^{\frac{1}{3}}

\frac{( t - 3 )}{( t ^{2} + 7 )}

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