Solve ∫ (from 0 to frac{ pi) of { 4}}ln(sqrt{2}) wrt x

$$\int_{ 0 }^{ \frac{ \pi }{ 4 } } \ln ( \sqrt{ 2 } ) d x$$

Evaluate

$$\frac{\pi \ln(2)}{8}\approx 0.272198261$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \log_{e}\left(\sqrt{2}\right)\mathrm{d}x$$

Find the integral of \log_{e}\left(\sqrt{2}\right) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\log_{e}\left(\sqrt{2}\right)x$$

Simplify.

$$\frac{\ln(2)x}{2}$$

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.

$$\frac{1}{2}\ln(2)\times \frac{1}{4}\pi -\frac{1}{2}\ln(2)\times 0$$

Simplify.

$$\frac{\pi \ln(2)}{8}$$