What is the integral of (4x^7-5x^2+2)/(x^7) ?

The integral of (4x^7-5x^2+2)/(x^7) is 4x+5/(4x^4)-1/(3x^6)+C

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integral of (4x^7-5x^2+2)/(x^7)

\int \frac{4 x ^{7} - 5 x ^{2} + 2}{x ^{7}} d x

4 x + \frac{5}{4 x ^{4}} - \frac{1}{3 x ^{6}} + C

Solution steps

\int \frac{4 x ^{7} - 5 x ^{2} + 2}{x ^{7}} d x

Expand

\frac{4 x ^{7} - 5 x ^{2} + 2}{x ^{7}} : 4 - \frac{5}{x ^{5}} + \frac{2}{x ^{7}}

= \int 4 - \frac{5}{x ^{5}} + \frac{2}{x ^{7}} 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 4 d x - \int \frac{5}{x ^{5}} d x + \int \frac{2}{x ^{7}} d x

\int 4 d x = 4 x

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

\int \frac{2}{x ^{7}} d x = - \frac{1}{3 x ^{6}}

= 4 x - (- \frac{5}{4 x ^{4}}) - \frac{1}{3 x ^{6}}

Simplify

= 4 x + \frac{5}{4 x ^{4}} - \frac{1}{3 x ^{6}}

Add a constant to the solution

= 4 x + \frac{5}{4 x ^{4}} - \frac{1}{3 x ^{6}} + C

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What is the integral of (4x^7-5x^2+2)/(x^7) ?
The integral of (4x^7-5x^2+2)/(x^7) is 4x+5/(4x^4)-1/(3x^6)+C