∫ (from 0 to 100) of (x^2-7x+9) wrt x-এর সমাধান করুন

$$\int _ { 0 } ^ { 100 } ( x ^ { 2 } - 7 x + 9 ) d x$$

মূল্যায়ন করুন

$$\frac{897700}{3}\approx 299233.333333333$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

$$\int x^{2}-7x+9\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -7x\mathrm{d}x+\int 9\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-7\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-7\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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}. -7 কে \frac{x^{2}}{2} বার গুণ করুন।

$$\frac{x^{3}}{3}-\frac{7x^{2}}{2}+\int 9\mathrm{d}x$$

Find the integral of 9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}-\frac{7x^{2}}{2}+9x$$

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.

$$\frac{100^{3}}{3}-\frac{7}{2}\times 100^{2}+9\times 100-\left(\frac{0^{3}}{3}-\frac{7}{2}\times 0^{2}+9\times 0\right)$$

সিমপ্লিফাই।

$$\frac{897700}{3}$$