Definite integral problem

**Dana_Scully** · January 19th 2009, 12:32 PM

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

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

**mr fantastic** · January 19th 2009, 12:37 PM

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 $\frac{d [\tan (a t)]}{dt} = a \, \sec^2 (at)$ .

**Danny** · January 19th 2009, 12:39 PM

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.

**Dana_Scully** · January 19th 2009, 12:58 PM

Thank you so much, I've got it now!

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