∫ (from 1 to 4) of 5/sqrt[4]{x wrt x-এর সমাধান করুন

$$\int _ { 1 } ^ { 4 } \frac { 5 } { \sqrt[ 4 ] { x } } d x$$

মূল্যায়ন করুন

$$\frac{40\sqrt{2}-20}{3}\approx 12.189514165$$

সমাধানের ধাপগুলো

Evaluate the indefinite integral first.

$$\int \frac{5}{\sqrt[4]{x}}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$5\int \frac{1}{\sqrt[4]{x}}\mathrm{d}x$$

x^{-\frac{1}{4}} হিসেবে \frac{1}{\sqrt[4]{x}} পুনরায় লিখুন৷ Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{1}{4}}\mathrm{d}x with \frac{x^{\frac{3}{4}}}{\frac{3}{4}}. সিমপ্লিফাই।

$$\frac{20x^{\frac{3}{4}}}{3}$$

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.

$$\frac{20}{3}\times 4^{\frac{3}{4}}-\frac{20}{3}\times 1^{\frac{3}{4}}$$

সিমপ্লিফাই।

$$\frac{40\sqrt{2}-20}{3}$$