∫ (from 3 to 4) of (x^2+5x+2) wrt x-এর সমাধান করুন

$$\int _ { 3 } ^ { 4 } ( x ^ { 2 } + 5 x + 2 ) d x$$

মূল্যায়ন করুন

$$\frac{191}{6}\approx 31.833333333$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

$$\int x^{2}+5x+2\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 5x\mathrm{d}x+\int 2\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x+5\int x\mathrm{d}x+\int 2\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+5\int x\mathrm{d}x+\int 2\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. 5 কে \frac{x^{2}}{2} বার গুণ করুন।

$$\frac{x^{3}}{3}+\frac{5x^{2}}{2}+\int 2\mathrm{d}x$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}+\frac{5x^{2}}{2}+2x$$

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.

$$\frac{4^{3}}{3}+\frac{5}{2}\times 4^{2}+2\times 4-\left(\frac{3^{3}}{3}+\frac{5}{2}\times 3^{2}+2\times 3\right)$$

সিমপ্লিফাই।

$$\frac{191}{6}$$