∫ (from 0 to 6) of (x^3-6x^2) wrt x-এর সমাধান করুন

$$\int _ { 0 } ^ { 6 } ( x ^ { 3 } - 6 x ^ { 2 } ) d x$$

মূল্যায়ন করুন

$$-108$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

$$\int x^{3}-6x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{3}\mathrm{d}x+\int -6x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{3}\mathrm{d}x-6\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$\frac{x^{4}}{4}-6\int x^{2}\mathrm{d}x$$

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}. -6 কে \frac{x^{3}}{3} বার গুণ করুন।

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

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.

$$\frac{6^{4}}{4}-2\times 6^{3}-\left(\frac{0^{4}}{4}-2\times 0^{3}\right)$$

সিমপ্লিফাই।

$$-108$$