∫ (from 0 to 8) of x^2+sinx wrt x eching

$$\int _ { 0 } ^ { 8 } x ^ { 2 } + \sin x d x$$

Baholash

$$\frac{515-3\cos(8)}{3}\approx 171,8121667$$

Yechim qadamlari

Evaluate the indefinite integral first.

$$\int x^{2}+\sin(x)\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int \sin(x)\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int \sin(x)\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result.

$$\frac{x^{3}}{3}-\cos(x)$$

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.

$$\frac{8^{3}}{3}-\cos(8)-\left(\frac{0^{3}}{3}-\cos(0)\right)$$

Qisqartirish.

$$\frac{1}{3}\left(-3\cos(8)+515\right)$$