Solve ∫ (from 5 to 15) of sqrt{(5e^2t)^2}

$$\int _ { 5 } ^ { 15 } \sqrt { ( 5 e ^ { 2 t } ) ^ { 2 } }$$

Evaluate

$$50e^{2t}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{\left(5e^{2t}\right)^{2}}\mathrm{d}x$$

Find the integral of \sqrt{\left(5e^{2t}\right)^{2}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\sqrt{\left(5e^{2t}\right)^{2}}x$$

Simplify.

$$5e^{2t}x$$

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.

$$5e^{2t}\times 15-5e^{2t}\times 5$$

Simplify.

$$50e^{2t}$$

Differentiate w.r.t. t

$$100e^{2t}$$