Solve ∫ (from 0 to pi) of sqrt{2x^3 wrt x

$$\int_{ 0 }^{ \pi } \sqrt{ 2 { x }^{ 3 } } d x$$

Evaluate

$$\frac{2\sqrt{2}\pi ^{\frac{5}{2}}}{5}\approx 9.89577178$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{2x^{3}}\mathrm{d}x$$

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

$$\sqrt{2}\int \sqrt{x^{3}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{3}{2}}\mathrm{d}x with \frac{2x^{\frac{5}{2}}}{5}.

$$\sqrt{2}\times \frac{2x^{\frac{5}{2}}}{5}$$

Simplify.

$$\frac{2\sqrt{2}x^{\frac{5}{2}}}{5}$$

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.

$$\frac{2}{5}\times 2^{\frac{1}{2}}\pi ^{\frac{5}{2}}-\frac{2}{5}\times 2^{\frac{1}{2}}\times 0^{\frac{5}{2}}$$

Simplify.

$$\frac{\left(2\pi \right)^{\frac{5}{2}}}{10}$$