Solve ∫ (from 0 to sin x) of sqrt{t}dt

$$\int _ { 0 } ^ { \sin x } \sqrt { t } d t$$

Evaluate

$$\frac{2\left(\sin(x)\right)^{\frac{3}{2}}}{3}$$

$$\exists n_{1}\in \mathrm{Z}\text{ : }\left(x\geq 2\pi n_{1}\text{ and }x\leq 2\pi n_{1}+\pi \right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{t}\mathrm{d}t$$

Rewrite \sqrt{t} as t^{\frac{1}{2}}. Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{\frac{1}{2}}\mathrm{d}t with \frac{t^{\frac{3}{2}}}{\frac{3}{2}}. Simplify.

$$\frac{2t^{\frac{3}{2}}}{3}$$

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.

$$\frac{2}{3}\left(\sin(x)\right)^{\frac{3}{2}}-\frac{2}{3}\times 0^{\frac{3}{2}}$$

Simplify.

$$\frac{2\left(\sin(x)\right)^{\frac{3}{2}}}{3}$$

Differentiate w.r.t. x

$$\sqrt{\sin(x)}\cos(x)$$