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When integrating a long chain of functions, we can integrate each term seperately and combine them. Let us now integrate:∫21x⁶dx = 21∫x⁶dx. Using the Power Rule [∫x^adx = (...

Write as y=uv u=x v=e^x

du/dx=1 dv/dx=e^x 

Using the product rule, dy/dx=vdu/dx + udv/dx

So dy/dx=e^x(1)+x(e^x...

So we have the equation initially in the form 3^(5x-2)=4^(6-x), and as the solution involves log10, then  a sensible first move would be to take log10 of both sides, giving log10(3^(5x-2)) = log10(4^(6...

Starting on the left hand side we can expand out the brackets to get:
(sinx + cosx)(sinx + cosx)
sin²x+sinxcosx+sinxcosx+cos²x
Grouping together the like terms we can ...

Tricky. Definitely can't do it by inspection (we don't know any fuction that just differentaties to ln(x)), it's not like we've really got anything to substitute u for if we wanted to do it by substitu...