Solve ∫ (from 1 to 2) of pi(1/sqrt{x})^2 wrt x

$$\int _ { 1 } ^ { 2 } \pi ( \frac { 1 } { \sqrt { x } } ) ^ { 2 } d x$$

Evaluate

$$\pi \ln(2)\approx 2.17758609$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \pi \times \left(\frac{1}{\sqrt{x}}\right)^{2}\mathrm{d}x$$

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

$$\pi \int \frac{1}{x}\mathrm{d}x$$

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

$$\pi \ln(|x|)$$

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.

$$\pi \ln(2)-\pi \ln(1)$$

Simplify.

$$\pi \ln(2)$$