What is the integral of 6sin(sqrt(x)) ?

The integral of 6sin(sqrt(x)) is 12(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C

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integral of 6sin(sqrt(x))

\int 6 sin (x) d x

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

Solution steps

\int 6 sin (x) d x

Take the constant out:

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

= 6 \cdot \int sin (x) d x

Apply u-substitution

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

= 6 \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

= 6 \cdot 2 \cdot \int sin (u) u d u

Apply Integration By Parts

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

= 6 \cdot 2 (- u cos (u) - \int - cos (u) d u)

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

= 6 \cdot 2 (- u cos (u) - (- sin (u)))

Substitute back

u = x

= 6 \cdot 2 (- x cos (x) - (- sin (x)))

Simplify

= 12 (- x cos (x) + sin (x))

Add a constant to the solution

= 12 (- x cos (x) + sin (x)) + C

\frac{d}{d t} (\frac{2 t}{2 + t})

e^{2 x} sin (3 x)

x y^{^{''}} = y^{^{'}} + (y^{^{'}})^{2}

\int \frac{9 + x}{x ^{2}} d x

f (x) = - 3 - 7 x^{2}, at x = 42

What is the integral of 6sin(sqrt(x)) ?
The integral of 6sin(sqrt(x)) is 12(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C