Solve ∫ (from 1 to sin x) of (3t^2)dt=

$$\int _ { 1 } ^ { \sin x } ( 3 t ^ { 2 } ) d t =$$

Evaluate

$$\left(\sin(x)\right)^{3}-1$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3t^{2}\mathrm{d}t$$

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

$$3\int t^{2}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}. Multiply 3 times \frac{t^{3}}{3}.

$$t^{3}$$

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.

$$\left(\sin(x)\right)^{3}-1^{3}$$

Simplify.

$$\left(\sin(x)\right)^{3}-1$$

Differentiate w.r.t. x

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