What is the integral of (xcos(x)+sin(x))sqrt(1+u) ?

The integral of (xcos(x)+sin(x))sqrt(1+u) is xsin(x)sqrt(1+u)+C

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integral of (xcos(x)+sin(x))sqrt(1+u)

\int (x cos (x) + sin (x)) 1 + u d x

x sin (x) 1 + u + C

Solution steps

\int (x cos (x) + sin (x)) 1 + u d x

Take the constant out:

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

= 1 + u \cdot \int sin (x) + cos (x) x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= 1 + u (- cos (x) + x sin (x) + cos (x))

Simplify

= x sin (x) 1 + u

Add a constant to the solution

= x sin (x) 1 + u + C

What is the integral of (xcos(x)+sin(x))sqrt(1+u) ?
The integral of (xcos(x)+sin(x))sqrt(1+u) is xsin(x)sqrt(1+u)+C