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

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

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

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

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

Solution steps

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

Long division

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

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

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

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

Add a constant to the solution

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

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