Solve ∫ (from 0 to 1) of (1000*6^t)dt

$$\int_{ 0 }^{ 1 } (1000 \times { 6 }^{ t } ) d t$$

Evaluate

$$\frac{5000}{\ln(6)}\approx 2790.553132756$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1000\times 6^{t}\mathrm{d}t$$

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

$$1000\int 6^{t}\mathrm{d}t$$

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

$$1000\times \frac{6^{t}}{\ln(6)}$$

Simplify.

$$\frac{1000\times 6^{t}}{\ln(6)}$$

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.

$$1000\times 6^{1}\ln(6)^{-1}-1000\times 6^{0}\ln(6)^{-1}$$

Simplify.

$$\frac{5000}{\ln(6)}$$