What is the integral of 1/4 cos(sqrt(x)) ?

The integral of 1/4 cos(sqrt(x)) is 1/2 (sqrt(x)sin(sqrt(x))+cos(sqrt(x)))+C

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integral of 1/4 cos(sqrt(x))

\int \frac{1}{4} cos (x) d x

\frac{1}{2} (x sin (x) + cos (x)) + C

Solution steps

\int \frac{1}{4} cos (x) d x

Take the constant out:

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

= \frac{1}{4} \cdot \int cos (x) d x

Apply u-substitution

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

= \frac{1}{4} \cdot \int cos (u) \cdot 2 u d u

Take the constant out:

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

= \frac{1}{4} \cdot 2 \cdot \int cos (u) u d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{4} \cdot 2 (u sin (u) - \int sin (u) d u)

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

= \frac{1}{4} \cdot 2 (u sin (u) - (- cos (u)))

Substitute back

u = x

= \frac{1}{4} \cdot 2 (x sin (x) - (- cos (x)))

Simplify

\frac{1}{4} \cdot 2 (x sin (x) - (- cos (x))) : \frac{1}{2} (x sin (x) + cos (x))

= \frac{1}{2} (x sin (x) + cos (x))

Add a constant to the solution

= \frac{1}{2} (x sin (x) + cos (x)) + C

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What is the integral of 1/4 cos(sqrt(x)) ?
The integral of 1/4 cos(sqrt(x)) is 1/2 (sqrt(x)sin(sqrt(x))+cos(sqrt(x)))+C