What is the integral of rcos(r^2) ?

The integral of rcos(r^2) is 1/2 sin(r^2)+C

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integral of rcos(r^2)

\int r cos (r^{2}) d r

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

Solution steps

\int r cos (r^{2}) d r

Apply u-substitution

= \int \frac{cos ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{2} sin (u)

Substitute back

u = r^{2}

= \frac{1}{2} sin (r^{2})

Add a constant to the solution

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