integral problem: int (sin(3t) 9sec^2(3t))/3 dt

degreeplus

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Oct 7, 2006
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I need help solving and understanding this integral

sin3t(9sec23t)3\displaystyle \int {\frac{{\sin 3t(9\sec ^2 3t)}}{3}}

the answer is -csc 3t but what I get is -sec 3t

Here is what I did:

I set the (9sec23t)\displaystyle (9\sec^2 3t) to 9cos23t\displaystyle \frac{{9}}{cos^2 3t}

then I multiplied it by sin 3t to get 3sin3tcos23t\displaystyle 3\int{\frac{{sin 3t}}{cos^2 3t}} and then turned it into 3sin3tcos3t1cos3t\displaystyle \int {3\frac{{\sin 3t}}{{\cos 3t}}} \bullet \frac{1}{{\cos 3t}}

then this becomes 3tan3tsec3t\displaystyle 3\int{tan 3t \bullet sec 3t}

and I get -sec 3t

Where did I go wrong?

Thanks for any help in advance :)
 
Re: integral problem

The solution is indeed sec(3t)\displaystyle sec(3t). You are correct, except for the negative.
 
degreeplus said:
I need help solving and understanding this integral

sin3t(9sec23t)3\displaystyle \int {\frac{{\sin 3t(9\sec ^2 3t)}}{3}}

the answer is -csc 3t but what I get is -sec 3t

Here is what I did:

I set the (9sec23t)\displaystyle (9\sec^2 3t) to 9cos23t\displaystyle \frac{{9}}{cos^2 3t}

then I multiplied it by sin 3t to get 3sin3tcos23t\displaystyle 3\int{\frac{{sin 3t}}{cos^2 3t}}

you could do substitution here (coming to same answer - except for the sign as galactus pointed out)

u = cos(t)

du = - sin(t) dt

Then integrand turns into -du/u^2

whose antiderivative is (1/u + C = ) sec(t) + C




and then turned it into 3sin3tcos3t1cos3t\displaystyle \int {3\frac{{\sin 3t}}{{\cos 3t}}} \bullet \frac{1}{{\cos 3t}}

then this becomes 3tan3tsec3t\displaystyle 3\int{tan 3t \bullet sec 3t}

and I get -sec 3t

Where did I go wrong?

Thanks for any help in advance :)
 
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