What is the integral of x(cos(x^2)+e^x) ?

The integral of x(cos(x^2)+e^x) is 1/2 sin(x^2)+e^xx-e^x+C

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integral of x(cos(x^2)+e^x)

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

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

Solution steps

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

Expand

x (cos (x^{2}) + e^{x}) : x cos (x^{2}) + e^{x} x

= \int x cos (x^{2}) + e^{x} x 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}) d x + \int e^{x} x d x

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

\int e^{x} x d x = e^{x} x - e^{x}

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

Add a constant to the solution

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

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What is the integral of x(cos(x^2)+e^x) ?
The integral of x(cos(x^2)+e^x) is 1/2 sin(x^2)+e^xx-e^x+C