Solve ∫ (from 0 to 2) of int_{y/2}^1cosx^2 wrt xdydz

$$\int _ { 0 } ^ { 2 } \int _ { \frac { y } { 2 } } ^ { 1 } \cos x ^ { 2 } d x d y d z$$

Evaluate

$$dx\left(2-y\right)\cos(x^{2})$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{2}\cos(x^{2})dx\mathrm{d}y\mathrm{d}z$$

Find the integral of \int _{0}^{2}\cos(x^{2})dx\mathrm{d}y using the table of common integrals rule \int a\mathrm{d}z=az.

$$\int _{0}^{2}\cos(x^{2})dx\mathrm{d}yz$$

Simplify.

$$2\cos(x^{2})dxz$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$2\cos(x^{2})dx-\frac{1}{2}\times 2\cos(x^{2})dxy$$

Simplify.

$$xd\cos(x^{2})\left(2-y\right)$$

Differentiate w.r.t. x

$$d\left(2-y\right)\left(-2x^{2}\sin(x^{2})+\cos(x^{2})\right)$$