Solve f ( x ) = ∫ _{x^2}^x^3e^2xdt

$$f ( x ) = \int _ { x ^ { 2 } } ^ { x ^ { 3 } } e ^ { 2 x } d t$$

Evaluate

$$\left(x-1\right)x^{2}e^{2x}$$

Solution Steps

Evaluate the indefinite integral first.

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

Find the integral of e^{2x} using the table of common integrals rule \int a\mathrm{d}t=at.

$$e^{2x}t$$

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.

$$e^{2x}x^{3}-e^{2x}x^{2}$$

Simplify.

$$\left(-1+x\right)x^{2}e^{2x}$$

Differentiate w.r.t. x

$$x\left(2x^{2}+x-2\right)e^{2x}$$