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Can I use change of variables to turn $\int_{3\pi/2}^{2\pi}\cos x\, dx$ into $\int_{0}^{\pi/2}\cos x\, dx$? Letting $x \mapsto x - 3\pi/2$ doesn't seem to work.

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It is dangerous to make a substitution and keep the variable name. –  André Nicolas Mar 18 '12 at 0:51
    
why do you even need to change variable in the first place. It is a simple integral –  Siddhi V Iyer Mar 18 '12 at 2:48
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2 Answers

up vote 2 down vote accepted

Try $u=2\pi-x$. The differential will change sign, but you’ll also have to turn the integral upside down.

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Simply you can't do any changes of variable to change the intervales, the primitive of $\cos x$ is $\sin x$ so why do you need this change of variable ! so

$$\int_{3\pi/2}^{2\pi}\cos x\, dx= [\sin x]_{3\pi/2}^{2\pi}=\sin 2\pi -\sin 3\pi/2 =0-(-1)=1$$

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