What is the integral of 6xcos(2x+5) ?

The integral of 6xcos(2x+5) is 6(1/2 xsin(2x+5)+1/4 cos(2x+5))+C

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integral of 6xcos(2x+5)

\int 6 x cos (2 x + 5) d x

6 (\frac{1}{2} x sin (2 x + 5) + \frac{1}{4} cos (2 x + 5)) + C

Solution steps

\int 6 x cos (2 x + 5) d x

Take the constant out:

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

= 6 \cdot \int x cos (2 x + 5) d x

Apply Integration By Parts

: \frac{1}{2} x sin (2 x + 5) - \int \frac{1}{2} sin (2 x + 5) d x

= 6 (\frac{1}{2} x sin (2 x + 5) - \int \frac{1}{2} sin (2 x + 5) d x)

\int \frac{1}{2} sin (2 x + 5) d x = - \frac{1}{4} cos (2 x + 5)

= 6 (\frac{1}{2} x sin (2 x + 5) - (- \frac{1}{4} cos (2 x + 5)))

Simplify

= 6 (\frac{1}{2} x sin (2 x + 5) + \frac{1}{4} cos (2 x + 5))

Add a constant to the solution

= 6 (\frac{1}{2} x sin (2 x + 5) + \frac{1}{4} cos (2 x + 5)) + C

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What is the integral of 6xcos(2x+5) ?
The integral of 6xcos(2x+5) is 6(1/2 xsin(2x+5)+1/4 cos(2x+5))+C