Լուծեք ∫ (from 0 to sqrt { 2) of / 2 } (1/2x^2-x^4) wrt x

$$\int _ { 0 } ^ { \frac { \sqrt { 2 } } { 2 } } ( \frac { 1 } { 2 } x ^ { 2 } - x ^ { 4 } ) d x$$

Գնահատել

$$\frac{\sqrt{2}}{60}\approx 0.023570226$$

Լուծման քայլերը

Evaluate the indefinite integral first.

$$\int \frac{x^{2}}{2}-x^{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{x^{2}}{2}\mathrm{d}x+\int -x^{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{\int x^{2}\mathrm{d}x}{2}-\int x^{4}\mathrm{d}x$$

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

$$\frac{x^{3}}{6}-\int x^{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Բազմապատկեք -1 անգամ \frac{x^{5}}{5}:

$$\frac{x^{3}}{6}-\frac{x^{5}}{5}$$

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.

$$\frac{1}{6}\times \left(\frac{1}{2}\times 2^{\frac{1}{2}}\right)^{3}-\frac{1}{5}\times \left(\frac{1}{2}\times 2^{\frac{1}{2}}\right)^{5}-\left(\frac{0^{3}}{6}-\frac{0^{5}}{5}\right)$$

Պարզեցնել:

$$\frac{\sqrt{2}}{60}$$