Solve ∫ (from 1 to 9) of 1/2tdt

$$\int _ { 1 } ^ { 9 } \frac { 1 } { 2 t } d t$$

Evaluate

$$\ln(3)\approx 1.098612289$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{2t}\mathrm{d}t$$

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

$$\frac{\int \frac{1}{t}\mathrm{d}t}{2}$$

Use \int \frac{1}{t}\mathrm{d}t=\ln(|t|) from the table of common integrals to obtain the result.

$$\frac{\ln(|t|)}{2}$$

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.

$$\frac{1}{2}\ln(|9|)-\frac{1}{2}\ln(|1|)$$

Simplify.

$$\ln(3)$$