Solve ∫ (from pi / 4 to pi / 2) of [4sinθ]_x=0^k=cscθdθ

$$\int _ { \pi / 4 } ^ { \pi / 2 } [ 4 \sin \theta ] _ { x = 0 } ^ { k = \csc \theta } d \theta$$

Evaluate

$$\pi \sin(\theta )$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4\sin(\theta )\mathrm{d}x$$

Find the integral of 4\sin(\theta ) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$4\sin(\theta )x$$

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.

$$\frac{1}{2}\times 4\sin(\theta )\pi -\frac{1}{4}\times 4\sin(\theta )\pi$$

Simplify.

$$\sin(\theta )\pi$$

Differentiate w.r.t. θ

$$\pi \cos(\theta )$$