What is the integral of sin(sqrt(at)) ?

The integral of sin(sqrt(at)) is 2/a (-cos(sqrt(at))sqrt(at)+sin(sqrt(at)))+C

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integral of sin(sqrt(at))

\int sin (a t) d t

\frac{2}{a} (- cos (a t) a t + sin (a t)) + C

Solution steps

\int sin (a t) d t

Apply u-substitution

: \int \frac{2 u sin ( u )}{a} d u

= \int \frac{2 u sin ( u )}{a} d u

Take the constant out:

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

= \frac{2}{a} \cdot \int u sin (u) d u

Apply Integration By Parts

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

= \frac{2}{a} (- u cos (u) - \int - cos (u) d u)

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

= \frac{2}{a} (- u cos (u) - (- sin (u)))

Substitute back

u = a t

= \frac{2}{a} (- a t cos (a t) - (- sin (a t)))

Simplify

= \frac{2}{a} (- cos (a t) a t + sin (a t))

Add a constant to the solution

= \frac{2}{a} (- cos (a t) a t + sin (a t)) + C

\frac{d}{d x} (e^{3 x} + 10 e^{2 x})

t^{2} y^{^{'}} + 3 t y + 2 t^{3} = 0

(e^{y} + 6 x) d x + (x e^{y}) d y = 0

\int e^{l n (2 t)} d t

\frac{1}{( s + 1 ) ^{2} ( s + 2 )}

What is the integral of sin(sqrt(at)) ?
The integral of sin(sqrt(at)) is 2/a (-cos(sqrt(at))sqrt(at)+sin(sqrt(at)))+C