`int sin(sqrt(theta)) / sqrt(theta) d theta` Find or evaluate the integral

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`int (sin sqrt theta)/sqrt theta d theta`

To solve, apply u-substitution method.

`u=sqrt theta`

`u= theta ^(1/2)`

`du = 1/2 theta^(-1/2) d theta`

`du = 1/(2theta^(1/2))d theta`

`du =1/(2 sqrt theta) d theta`

`2du =1/sqrt theta d theta`

Expressing the integral in terms of u, it becomes:

`= int sin...

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`int (sin sqrt theta)/sqrt theta d theta`

To solve, apply u-substitution method.

`u=sqrt theta`

`u= theta ^(1/2)`

`du = 1/2 theta^(-1/2) d theta`

`du = 1/(2theta^(1/2))d theta`

`du =1/(2 sqrt theta) d theta`

`2du =1/sqrt theta d theta`

Expressing the integral in terms of u, it becomes:

`= int sin (sqrt theta) * 1/sqrt theta d theta`

`= int sin (u) * 2du`

`= 2 int sin (u) du`

Then, apply the integral formula `int sin (x) dx = -cos(x) + C` .

`= 2*(-cos (u)) + C`

`= -2cos(u) + C`

And, substitute back  `u = sqrt theta` .

`= -2cos( sqrt theta) + C`

 

Therefore, `int (sin sqrt theta)/sqrt theta d theta= -2cos( sqrt theta) + C` .

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