What is the integral of 8sin(sqrt(x)) ?

The integral of 8sin(sqrt(x)) is 16(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C

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integral of 8sin(sqrt(x))

\int 8 sin (x) d x

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

Solution steps

\int 8 sin (x) d x

Take the constant out:

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

= 8 \cdot \int sin (x) d x

Apply u-substitution

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

= 8 \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

= 8 \cdot 2 \cdot \int sin (u) u d u

Apply Integration By Parts

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

= 8 \cdot 2 (- u cos (u) - \int - cos (u) d u)

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

= 8 \cdot 2 (- u cos (u) - (- sin (u)))

Substitute back

u = x

= 8 \cdot 2 (- x cos (x) - (- sin (x)))

Simplify

= 16 (- x cos (x) + sin (x))

Add a constant to the solution

= 16 (- x cos (x) + sin (x)) + C

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\frac{d y}{d x} = - y^{2} sin (x)

\int \frac{1}{x ( x ^{2} - 49 ) ^{\frac{3}{2}}} d x

\frac{d}{d x} (x \cdot ln (x) - 1)

\frac{d}{d x} (\frac{tan ( x ) - 2}{sec ( x )})

What is the integral of 8sin(sqrt(x)) ?
The integral of 8sin(sqrt(x)) is 16(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C