$\displaystyle \int x(1-2e^{cotx^{2}})csc^{2}(x^{2})dx$
$\displaystyle \int5 e^{\frac{1}{2}lnx}dx$
$\displaystyle \int ln e^{\frac{x}{2}}dx$
Can you show how it's done?
Thank you
$\displaystyle \int x(1-2e^{cotx^{2}})csc^{2}(x^{2})dx$
$\displaystyle \int5 e^{\frac{1}{2}lnx}dx$
$\displaystyle \int ln e^{\frac{x}{2}}dx$
Can you show how it's done?
Thank you
Expand the brackets and tackle 2 separate integrals.
Just in case a picture helps...
As usual, straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed lines similarly but with respect to the dashed balloon expression. So the triangular network...
... is the chain rule.
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D'oh! Cheers for that... and if you don't mind another picture...
... where we're doing just as you say, expressing $\displaystyle e^{\cot(x^2)}$ in terms of u (i.e. the dashed balloon expression)...
... in order to divide by $\displaystyle e^{\cot(x^2)}$ and thus cancel the extra one appearing in the derivative by-product. I'm using F and G just to point up how the integration with respect to u splits into two.
Obviously, the lower equals sign depends on several lines of algebra that aren't shown, so the picture's only an overview.
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Don't integrate - balloontegrate!
Balloon Calculus Forum