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

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

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integral of (3x^4-2x^2+1)/(x^4)

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

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

Solution steps

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

Expand

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

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

\int 3 d x = 3 x

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

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

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

Simplify

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

Add a constant to the solution

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

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