Solve ∫ (from 0 to 1) of 1000*8^tdt

$$\int_{ 0 }^{ 1 } 1000 \times { 8 }^{ t } d t$$

Evaluate

$$\frac{7000}{3\ln(2)}\approx 3366.288428741$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1000\times 8^{t}\mathrm{d}t$$

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

$$1000\int 8^{t}\mathrm{d}t$$

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

$$\frac{8^{t}}{\ln(8)}$$

Simplify.

$$\frac{1000\times \frac{8^{t}}{\ln(2)}}{3}$$

Simplify.

$$\frac{1000\times 8^{t}}{3\ln(2)}$$

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.

$$\frac{1000}{3}\times 8^{1}\ln(2)^{-1}-\frac{1000}{3}\times 8^{0}\ln(2)^{-1}$$

Simplify.

$$\frac{7000}{3\ln(2)}$$