Solve ∫ (from 0 to 6) of 1425(1.12^t)dt

$$\int _ { 0 } ^ { 6 } 1425 ( 1.12 ^ { t } ) d t$$

Evaluate

$$\frac{13551731703}{9765625\ln(1.12)}\approx 12244.890361412$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1425\times 1.12^{t}\mathrm{d}t$$

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

$$1425\int 1.12^{t}\mathrm{d}t$$

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

$$\frac{1.12^{t}}{\ln(1.12)}$$

Simplify.

$$1425\times \frac{1.12^{t}}{\ln(\frac{28}{25})}$$

Simplify.

$$\frac{1425\times 1.12^{t}}{\ln(\frac{28}{25})}$$

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.

$$1425\times 1.12^{6}\left(\ln(28)-2\ln(5)\right)^{-1}-1425\times 1.12^{0}\left(\ln(28)-2\ln(5)\right)^{-1}$$

Simplify.

$$\frac{13551731703}{9765625\ln(\frac{28}{25})}$$