What is the integral of sin(sqrt(x+4)) ?

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

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integral of sin(sqrt(x+4))

\int sin (x + 4) d x

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

Solution steps

\int sin (x + 4) d x

Apply u-substitution

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

= \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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x + 4

= 2 (- x + 4 cos (x + 4) - (- sin (x + 4)))

Simplify

= 2 (- cos (x + 4) x + 4 + sin (x + 4))

Add a constant to the solution

= 2 (- cos (x + 4) x + 4 + sin (x + 4)) + C

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

x \to - 1 - lim (x)

f (x) = 5 arcsin (x^{4})

\frac{d}{d x} (1 - 25 x^{2} arccos (5 x))

x^{^{'}} = 2 (x + 1), x (0) = 0

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