Calculus Final Tomarrow And I need help?

Question

I'm wondering if anyone can help me with integrals and their applications. I just can't get the whole u substitution thing down. I know it undoes the chain rule but I don't know how to apply it. Also need help with derivatives and antiderivatives of exponentials and logarithmic functions. If anyone took some time to try and clear this stuff up for me I'd be eternally grateful.

Answer 1

integral 2x(x^2+1)^3 dx
Let u = x^2+1
Then du/dx= 2x --> du = 2x dx
So replace x^2+1 with u and 2xdx with du and get
integral u^3 du = (1/4)u^4 +C =(1/4) (x^2+1)^4 +C.

integral x*sqrt(1+x^2) dx
Let u = 1+x^2
Then du/dx = 2x --> du = 2xdx
So integral sqrt(1+x^2) xdx = 1/2integ sqrt(1+x^2)^1/2 (2xdx)
Then integral 1/2 u^1/2du = (1/2 u^3/2)/3/2 + C
= 1/3((1+x^2)^3/2 + C

Remember that dy/dx (ln u) = 1/u du/dx, so if u =2x+1,
then dy/dx ln(2x+1) = 1/(2x+1) *2 = 2/2x+1)

dy/dx a^u = a^u ln a du/dx, so
dy/dx 2^(3x^2+1) = 2^(3x^2+1) ln2 * 6x

I hope that helped a bit.

Answer 2

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