Solve ∫ (from - 2 to 3) of 9t^2+72^t+4

$$\int _ { - 2 } ^ { 3 } 9 t ^ { 2 } + 72 ^ { t } + 4$$

Evaluate

$$5\left(9t^{2}+72^{t}+4\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 9t^{2}+72^{t}+4\mathrm{d}x$$

Find the integral of 9t^{2}+72^{t}+4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(9t^{2}+72^{t}+4\right)x$$

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.

$$\left(9t^{2}+72^{t}+4\right)\times 3-\left(9t^{2}+72^{t}+4\right)\left(-2\right)$$

Simplify.

$$45t^{2}+5\times 72^{t}+20$$

Differentiate w.r.t. t

$$5\left(\ln(72)\times 72^{t}+18t\right)$$