# Definite integral problem

• January 19th 2009, 12:32 PM
Dana_Scully
Definite integral problem
Could someone show me the steps for a problem like this? I missed a class and you'd be saving my life.

$\displaystyle\int^{\pi}_0 \sec^2 \left(\frac{t}{3} \right) dt$
• January 19th 2009, 12:37 PM
mr fantastic
Quote:

Originally Posted by Dana_Scully
Could someone show me the steps for a problem like this? I missed a class and you'd be saving my life.

$\displaystyle\int^{\pi}_0 \sec^2 \left(\frac{t}{3}\right) dt$

Note that $\frac{d [\tan (a t)]}{dt} = a \, \sec^2 (at)$.
• January 19th 2009, 12:39 PM
Jester
Quote:

Originally Posted by Dana_Scully
Could someone show me the steps for a problem like this? I missed a class and you'd be saving my life.

$\displaystyle\int^{\pi}_0 \sec ^2 \left(\frac{t}{3}\right) dt$

Make the substitution $u = \frac{t}{3}$ so $du = \frac{dt}{3}$. New limits of integration $t = 0 \; \Rightarrow \; u = 0, \; \; t = \pi \; \Rightarrow \; u = \frac{ \pi }{3}$

New problem

$3 \int_0^{\frac{\pi}{3}} \sec^2 u \,du$

You should recognize the antiderivative for this.
• January 19th 2009, 12:58 PM
Dana_Scully
Thank you so much, I've got it now!