Solve H ( x ) = ∫ (from - 5 to cos x) of 2t^2dt

$$H ( x ) = \int _ { - 5 } ^ { \cos x } 2 t ^ { 2 } d t$$

Evaluate

$$\frac{2\left(\left(\cos(x)\right)^{3}+125\right)}{3}$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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}.

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

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.

$$\frac{2}{3}\left(\cos(x)\right)^{3}-\frac{2}{3}\left(-5\right)^{3}$$

Simplify.

$$\frac{1}{3}\left(3\times \frac{2\left(\cos(x)\right)^{3}}{3}+250\right)$$

Differentiate w.r.t. x

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