Solve ∫ (from 0 to { x) of ^2}4xcos(y) wrt y

$$\int_{ 0 }^{ { x }^{ 2 } } 4x \cos ( y ) d y$$

Evaluate

$$4x\sin(x^{2})$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x\cos(y)\mathrm{d}y$$

Factor out the constant using \int af\left(y\right)\mathrm{d}y=a\int f\left(y\right)\mathrm{d}y.

$$4x\int \cos(y)\mathrm{d}y$$

Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.

$$4x\sin(y)$$

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.

$$4x\sin(x^{2})-4x\sin(0)$$

Simplify.

$$4x\sin(x^{2})$$

Differentiate w.r.t. x

$$4\left(2x^{2}\cos(x^{2})+\sin(x^{2})\right)$$