I need help with this integral

[natHenderson]

I don't even know where to start with this problem, all I've done is substitute in 5sinx for x, can somebody help?

 

[Deleted member 4993]

I don't even know where to start with this problem, all I've done is substitute in 5sinx for x, can somebody help?
View attachment 19042

You write:

"I've done is substitute in 5sinx for x" ....... you cannot do that!

However, you can substitute

5 sin(t) = x

Then,

What do you get for,

√( 25 - x²) = ?

 

[natHenderson]

I got 5cos(t)

 

[Deleted member 4993]

I got 5cos(t)

Correct - good work! Now, if

x = 5 * sin(t)

(dx)/(dt) = ? (remember you are doing calculus now)

 

[natHenderson]

dx/dt = 5cos(x)
So, then I wrote the 1/5 times the integral of 1/(sin^2(t)*5cos(t) * 5cos(x)dt and the 5cos(t) cancelled out which left me with 1/5 times the integral of 1/sin^2(t) dt which is equal to 1/5 * integral of csc^2(t)dt which gave me -1/5*cot(x), is that correct?

 

[natHenderson]

Actually, I didn't need to evaluate the integral, so my answer would be B, is that correct?

 

[HallsofIvy]

It might help you to recall that \(\displaystyle \frac{1}{cos(\theta)}= sec(\theta)\).

As Subhotosh Kahn said you cannot substitute "5 sin(x)" for x!

But since the offered possible answers involve \(\displaystyle \theta\) I can't imagine why you would not use
\(\displaystyle \theta\) as new variable rather than t or u or anything else.

 

[natHenderson]

Yes, when I initially wrote the post, I meant to say I substituted the x for 5sin(x), not the other way around, but thank you for your input!

 

[Deleted member 4993]

Actually, I didn't need to evaluate the integral, so my answer would be B, is that correct?

No!!

 

[natHenderson]

Oh, wait, I miscalculated, it's C right?