What is the integral of xcos(x^2+1)+x/(x^2+1) ?

The integral of xcos(x^2+1)+x/(x^2+1) is 1/2 sin(x^2+1)+1/2 ln|x^2+1|+C

integral of xcos(x^2+1)+x/(x^2+1)

\int x cos (x^{2} + 1) + \frac{x}{x ^{2} + 1} d x

\frac{1}{2} sin (x^{2} + 1) + \frac{1}{2} ln x^{2} + 1 + C

Solution steps

\int x cos (x^{2} + 1) + \frac{x}{x ^{2} + 1} 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 cos (x^{2} + 1) d x + \int \frac{x}{x ^{2} + 1} d x

\int x cos (x^{2} + 1) d x = \frac{1}{2} sin (x^{2} + 1)

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

= \frac{1}{2} sin (x^{2} + 1) + \frac{1}{2} ln x^{2} + 1

Add a constant to the solution

= \frac{1}{2} sin (x^{2} + 1) + \frac{1}{2} ln x^{2} + 1 + C

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What is the integral of xcos(x^2+1)+x/(x^2+1) ?
The integral of xcos(x^2+1)+x/(x^2+1) is 1/2 sin(x^2+1)+1/2 ln|x^2+1|+C