What is the integral of (xe^{x^2}-x/(x^2+7)) ?

The integral of (xe^{x^2}-x/(x^2+7)) is 1/2 e^{x^2}-1/2 ln|x^2+7|+C

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integral of (xe^{x^2}-x/(x^2+7))

\int (x e^{x^{2}} - \frac{x}{x ^{2} + 7}) d x

\frac{1}{2} e^{x^{2}} - \frac{1}{2} ln x^{2} + 7 + C

Solution steps

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

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

\int \frac{x}{x ^{2} + 7} d x = \frac{1}{2} ln x^{2} + 7

= \frac{1}{2} e^{x^{2}} - \frac{1}{2} ln x^{2} + 7

Add a constant to the solution

= \frac{1}{2} e^{x^{2}} - \frac{1}{2} ln x^{2} + 7 + C

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What is the integral of (xe^{x^2}-x/(x^2+7)) ?
The integral of (xe^{x^2}-x/(x^2+7)) is 1/2 e^{x^2}-1/2 ln|x^2+7|+C