Please Help Evaluate the integral

thatguy47

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Aug 11, 2008
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\(\displaystyle \int_\)From 0 to (pi/4) of 4 tan^3xdx

I pulled out the 4 so it was 4 times int(0 > pi/4) of tan^3xdx

I'm tried doing u-substitution but I couldn't figure it out. Please help.
 
Hello, thatguy47!

0π4 ⁣4tan3xdx\displaystyle \int_0^{\frac{\pi}{4}\!}4\tan^3x\,dx

We have: 4 ⁣0π4 ⁣tan2xtanxdx=4 ⁣0π4 ⁣(sec2 ⁣x1)tanxdx\displaystyle \text{We have: }\:4\!\int^{\frac{\pi}{4}}_0\! \tan^2x\,\tan x\,dx \:=\:4\!\int^{\frac{\pi}{4}}_0\!(\sec^2\!x - 1)\tan x\,dx

. . =  4 ⁣0π4 ⁣[tanxsec2 ⁣xtanx]dx  =  4[0π4 ⁣tanxsec2 ⁣xdx0π4 ⁣tanxdx]\displaystyle = \;4\!\int^{\frac{\pi}{4}}_0\!\bigg[\tan x\,\sec^2\!x - \tan x\bigg]\,dx \;=\;4\left[\int^{\frac{\pi}{4}}_0\!\tan x\,\sec^2\!x\,dx - \int^{\frac{\pi}{4}}_0\!\tan x\,dx\right]

Got it?

 
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