∫ (from - 2 to 5) of (4x^2-2x+1) wrt x-এর সমাধান করুন

$$\int _ { - 2 } ^ { 5 } ( 4 x ^ { 2 } - 2 x + 1 ) d x$$

মূল্যায়ন করুন

$$\frac{490}{3}\approx 163.333333333$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

$$\int 4x^{2}-2x+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{2}\mathrm{d}x+\int -2x\mathrm{d}x+\int 1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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}. 4 কে \frac{x^{3}}{3} বার গুণ করুন।

$$\frac{4x^{3}}{3}-2\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}-x^{2}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{4x^{3}}{3}-x^{2}+x$$

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.

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

সিমপ্লিফাই।

$$\frac{490}{3}$$