What is the integral of (sqrt(2x)+sin(3x)) ?

The integral of (sqrt(2x)+sin(3x)) is sqrt(2)xsqrt(x)-(sqrt(2))/3 x^{3/2}-1/3 cos(3x)+C

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integral of (sqrt(2x)+sin(3x))

\int (2 x + sin (3 x)) d x

2 x x - \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{3} cos (3 x) + C

Solution steps

\int 2 x + sin (3 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

= \int 2 x d x + \int sin (3 x) d x

\int 2 x d x = 2 x x - \frac{2}{3} x^{\frac{3}{2}}

\int sin (3 x) d x = - \frac{1}{3} cos (3 x)

= 2 x x - \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{3} cos (3 x)

Add a constant to the solution

= 2 x x - \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{3} cos (3 x) + C

What is the integral of (sqrt(2x)+sin(3x)) ?
The integral of (sqrt(2x)+sin(3x)) is sqrt(2)xsqrt(x)-(sqrt(2))/3 x^{3/2}-1/3 cos(3x)+C