Evaluate integrals: int [0,2] [(e^x - e^(-x))^2] dx, etc

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Sophie

Junior Member
Joined
Feb 7, 2007
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67
Hello

2 integrals I am supose to evaluate have got the better of me. I have been trying a couple of substitutions but I am getting no where. If someone could point towards a starting point I would be very greatful. Please please do not tell me how to do the whole thing as I learn best doing the examples myself, I just need a starting point.

\(\displaystyle \L\
\mbox{1. }\, \int\limits_0^2\, {\left( {e^x\, -\, e^{ - x} } \right)} ^2\, {\text{dx}}\)

\(\displaystyle \L\
\mbox{2. }\, \int\, {\frac{{\text{x}}}
{{\sqrt {{\text{1 - 2x}}} }}\, {\text{dx}}}\)

Thanks Sophie
 
For the first one, expand the bracket and integrate term by term. If you don't know how to integrate e^{+-ax} with respect to x (where a is a constant), you're in trouble.

For the second one, try making the substitution u = 1 - 2x --> du = -2dx.

Edit: upon closer inspection, you can recognize e^x - e^-x to be equal to -2i*sin(xi), but if you aren't familiar with imaginary numbers, don't bother.
 
?? You may wish to recognize \(\displaystyle \L\;e^{x}\;-\;e^{-x}\;=\;2sinh(x)\).
 
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