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First, you use the substitution u=1+e x which implies that du=e x dx. Then, you factorise e 3x = e 2x e x and replace e x dx with du. Then by rearranging, e 2x = (u-1) 2 , so you are now ready to substitute ...

First, use the substitution u=e^x (which implies dx=du/u) to make the integral ∫6/(u*(u+2)))du. Next seperate the fraction using partial fractions and expand to form 3∫1/u du - 3∫1/(u+2) du. Next integrate t...

This is a very cunning application of the integration by parts rule. Although it might look at first like integration by parts doesn't apply here since there is only the one factor, there is actually a hidde...

6 ∫ e^(2x+1) dx 6 ∫ (1/2)(e^(2x+1))3e^(2x+1) + c

36/x^2 = (36)^0.5/(x^2)^0.56/x = 1/6x