∫ (from 0 to 1) of 1/2sqrt{x^3 wrt x-এর সমাধান করুন

$$\int_{ 0 }^{ 1 } \frac{ 1 }{ 2 } \sqrt{ { x }^{ 3 } } d x$$

মূল্যায়ন করুন

$$\frac{1}{5}=0.2$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

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

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

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

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}.

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

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.

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

সিমপ্লিফাই।

$$\frac{1}{5}$$