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

The integral of 2/(sqrt(x))sin(sqrt(x)+2) is -4cos(2+sqrt(x))+C

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

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

- 4 cos (2 + x) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Expand

\frac{sin ( 2 ) cos ( x ) + cos ( 2 ) sin ( x )}{x} : \frac{sin ( 2 ) cos ( x )}{x} + \frac{cos ( 2 ) sin ( x )}{x}

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

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

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

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

= 2 (sin (2) \cdot 2 sin (x) - 2 cos (2) cos (x))

Simplify

2 (sin (2) \cdot 2 sin (x) - 2 cos (2) cos (x)) : - 4 cos (2 + x)

= - 4 cos (2 + x)

Add a constant to the solution

= - 4 cos (2 + x) + C

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

\int \frac{x - 1}{x ^{2} - 3 x + 1} d x

\int 3 x (x^{2} - 5)^{5} d x

\int (- 3 + \frac{12}{p}) d p

\int 4 sin (x^{2}) d x

What is the integral of 2/(sqrt(x))sin(sqrt(x)+2) ?
The integral of 2/(sqrt(x))sin(sqrt(x)+2) is -4cos(2+sqrt(x))+C