What is the integral of (x+5x^7)/(x^3) ?

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

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integral of (x+5x^7)/(x^3)

\int \frac{x + 5 x ^{7}}{x ^{3}} d x

- \frac{1}{x} + x^{5} + C

Solution steps

\int \frac{x + 5 x ^{7}}{x ^{3}} d x

Expand

\frac{x + 5 x ^{7}}{x ^{3}} : \frac{1}{x ^{2}} + 5 x^{4}

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

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

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

= - \frac{1}{x} + x^{5}

Add a constant to the solution

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

\frac{ln ( x )}{x ^{2}}

x ln (x)

\int \frac{1}{81 + 64 x ^{2}} d x

\int \frac{4 x ^{2}}{4 - 9 x ^{6}} d x

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

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