What is the integral of 2cos(sqrt(x)) ?

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

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integral of 2cos(sqrt(x))

\int 2 cos (x) d x

4 (x sin (x) + cos (x)) + C

Solution steps

\int 2 cos (x) d x

Take the constant out:

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

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

Apply u-substitution

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

= 2 \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

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

Apply Integration By Parts

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

= 2 \cdot 2 (u sin (u) - \int sin (u) d u)

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

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

Substitute back

u = x

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

Simplify

= 4 (x sin (x) + cos (x))

Add a constant to the solution

= 4 (x sin (x) + cos (x)) + C

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