Solve ∫ (from 2 to 3) of 3^xd

$$\int _ { 2 } ^ { 3 } 3 ^ { x } d$$

Evaluate

$$\frac{18d}{\ln(3)}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3^{x}d\mathrm{d}x$$

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

$$d\int 3^{x}\mathrm{d}x$$

Use \int x^{d}\mathrm{d}d=\frac{x^{d}}{\ln(x)} from the table of common integrals to obtain the result.

$$d\times \frac{3^{x}}{\ln(3)}$$

Simplify.

$$\frac{d\times 3^{x}}{\ln(3)}$$

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.

$$d\times 3^{3}\ln(3)^{-1}-d\times 3^{2}\ln(3)^{-1}$$

Simplify.

$$\frac{18d}{\ln(3)}$$

Differentiate w.r.t. d

$$\frac{18}{\ln(3)}$$