What is the integral of 1/(sqrt(x))cos(sqrt(x)+3) ?

The integral of 1/(sqrt(x))cos(sqrt(x)+3) is 2sin(sqrt(x)+3)+C

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integral of 1/(sqrt(x))cos(sqrt(x)+3)

\int \frac{1}{x} cos (x + 3) d x

2 sin (x + 3) + C

Solution steps

\int \frac{1}{x} cos (x + 3) d x

Simplify

\frac{1}{x} cos (x + 3) : \frac{cos ( x + 3 )}{x}

= \int \frac{cos ( x + 3 )}{x} d x

Apply u-substitution

: \int 2 cos (u) d u

= \int 2 cos (u) d u

Take the constant out:

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

= 2 \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= 2 sin (u)

Substitute back

u = x + 3

= 2 sin (x + 3)

Add a constant to the solution

= 2 sin (x + 3) + C

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What is the integral of 1/(sqrt(x))cos(sqrt(x)+3) ?
The integral of 1/(sqrt(x))cos(sqrt(x)+3) is 2sin(sqrt(x)+3)+C