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

The integral of 4sin(sqrt(x)) is 8(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C

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integral of 4sin(sqrt(x))

\int 4 sin (x) d x

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

Solution steps

\int 4 sin (x) d x

Take the constant out:

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

= 4 \cdot \int sin (x) d x

Apply u-substitution

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

= 4 \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

= 4 \cdot 2 \cdot \int sin (u) u d u

Apply Integration By Parts

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

= 4 \cdot 2 (- u cos (u) - \int - cos (u) d u)

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

= 4 \cdot 2 (- u cos (u) - (- sin (u)))

Substitute back

u = x

= 4 \cdot 2 (- x cos (x) - (- sin (x)))

Simplify

= 8 (- x cos (x) + sin (x))

Add a constant to the solution

= 8 (- x cos (x) + sin (x)) + C

\frac{d y}{d t} = 7 t e^{2 y}

\frac{\partial}{\partial x} (0.5 x arctan (\frac{y}{x}))

\int x^{2} sin (\frac{x}{2}) d x

\int \frac{1 + x}{x} d x

2 \frac{d y}{d x} - 4 x y = 8 x, y (0) = 2

What is the integral of 4sin(sqrt(x)) ?
The integral of 4sin(sqrt(x)) is 8(-sqrt(x)cos(sqrt(x))+sin(sqrt(x)))+C