∫ (from 0 to pi) of (cosx+x^2) wrt x-এর সমাধান করুন

$$\int _ { 0 } ^ { \pi } ( \cos x + x ^ { 2 } ) d x$$

মূল্যায়ন করুন

$$\frac{\pi ^{3}}{3}\approx 10.33542556$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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

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

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}.

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

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.

$$\sin(\pi )+\frac{\pi ^{3}}{3}-\left(\sin(0)+\frac{0^{3}}{3}\right)$$

সিমপ্লিফাই।

$$\frac{\pi ^{3}}{3}$$