What is the integral of 1/2 xsin(x+1) ?

The integral of 1/2 xsin(x+1) is 1/2 (-xcos(x+1)+sin(x+1))+C

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integral of 1/2 xsin(x+1)

\int \frac{1}{2} x sin (x + 1) d x

\frac{1}{2} (- x cos (x + 1) + sin (x + 1)) + C

Solution steps

\int \frac{1}{2} x sin (x + 1) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int x sin (x + 1) d x

Apply Integration By Parts

: - x cos (x + 1) - \int - cos (x + 1) d x

= \frac{1}{2} (- x cos (x + 1) - \int - cos (x + 1) d x)

\int - cos (x + 1) d x = - sin (x + 1)

= \frac{1}{2} (- x cos (x + 1) - (- sin (x + 1)))

Simplify

= \frac{1}{2} (- x cos (x + 1) + sin (x + 1))

Add a constant to the solution

= \frac{1}{2} (- x cos (x + 1) + sin (x + 1)) + C

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What is the integral of 1/2 xsin(x+1) ?
The integral of 1/2 xsin(x+1) is 1/2 (-xcos(x+1)+sin(x+1))+C