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

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

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

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

\frac{2}{3} sin (3 x) + C

Solution steps

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

Apply u-substitution

: \int \frac{2 cos ( u )}{3} d u

= \int \frac{2 cos ( u )}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{2}{3} sin (u)

Substitute back

u = 3 x

= \frac{2}{3} sin (3 x)

Add a constant to the solution

= \frac{2}{3} sin (3 x) + C

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

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

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

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

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

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