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

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

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integral of sin(sqrt(t))

\int sin (t) d t

2 (- t cos (t) + sin (t)) + C

Solution steps

\int sin (t) d t

Apply u-substitution

= \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

= 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 = t

= 2 (- t cos (t) - (- sin (t)))

Simplify

= 2 (- t cos (t) + sin (t))

Add a constant to the solution

= 2 (- t cos (t) + sin (t)) + C

\frac{d}{d x} (a \cdot x^{2} + b \cdot x + c)

\int \frac{x ^{3} + x + 1}{1 + x ^{2}} d x

(t) e^{4 t}

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

\int \frac{1}{9 x ^{2} - 49} d x

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