What is the integral of 5sin(sqrt(x)) ?

The integral of 5sin(sqrt(x)) is 10(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C

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integral of 5sin(sqrt(x))

\int 5 sin (x) d x

10 (- x cos (x) + sin (x)) + C

Solution steps

\int 5 sin (x) d x

Take the constant out:

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

= 5 \cdot \int sin (x) d x

Apply u-substitution

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

= 5 \cdot \int sin (u) \cdot 2 u d u

Take the constant out:

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

= 5 \cdot 2 \cdot \int sin (u) u d u

Apply Integration By Parts

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

= 5 \cdot 2 (- u cos (u) - \int - cos (u) d u)

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

= 5 \cdot 2 (- u cos (u) - (- sin (u)))

Substitute back

u = x

= 5 \cdot 2 (- x cos (x) - (- sin (x)))

Simplify

= 10 (- x cos (x) + sin (x))

Add a constant to the solution

= 10 (- x cos (x) + sin (x)) + C

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What is the integral of 5sin(sqrt(x)) ?
The integral of 5sin(sqrt(x)) is 10(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C