What is the integral of (1-sin(sqrt(x)))/(sqrt(x)) ?

The integral of (1-sin(sqrt(x)))/(sqrt(x)) is 2sqrt(x)+2cos(sqrt(x))+C

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integral of (1-sin(sqrt(x)))/(sqrt(x))

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

2 x + 2 cos (x) + C

Solution steps

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

Expand

\frac{1 - sin ( x )}{x} : \frac{1}{x} - \frac{sin ( x )}{x}

= \int \frac{1}{x} - \frac{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

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

\int \frac{1}{x} d x = 2 x

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

= 2 x - (- 2 cos (x))

Simplify

= 2 x + 2 cos (x)

Add a constant to the solution

= 2 x + 2 cos (x) + C

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