What is the antiderivative of $x sin (\frac{x}{2})$ $x sin (x/2)$ ?

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What is the antiderivative of $x sin (\frac{x}{2})$ $x sin (x/2)$ ?

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Answer 1 · 2018-05-17T23:48:43

$\int x sin (\frac{x}{2}) d x = - 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}) + C$ $int xsin(x/2)dx =-2xcos(x/2) + 4sin(x/2)+C$

Explanation:

Evaluate the indefinite integral:

$\int x sin (\frac{x}{2}) d x$ $int xsin(x/2)dx$

by parts:

$\int x sin (\frac{x}{2}) d x = \int x \frac{d}{d x} (- 2 cos (\frac{x}{2})) d x$ $int xsin(x/2)dx = int x d/dx (-2cos(x/2))dx$

$\int x sin (\frac{x}{2}) d x = - 2 x cos (\frac{x}{2}) + 2 \int cos (\frac{x}{2}) d x$ $int xsin(x/2)dx =-2xcos(x/2) + 2 int cos(x/2)dx$

$\int x sin (\frac{x}{2}) d x = - 2 x cos (\frac{x}{2}) + 4 \int cos (\frac{x}{2}) d (\frac{x}{2})$ $int xsin(x/2)dx =-2xcos(x/2) + 4 int cos(x/2)d(x/2)$

$\int x sin (\frac{x}{2}) d x = - 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}) + C$ $int xsin(x/2)dx =-2xcos(x/2) + 4sin(x/2)+C$

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What is the antiderivative of $x sin (\frac{x}{2})$ $x sin (x/2)$ ?

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