Indefinite/Definite Integrals

Question

I need help solving a couple of indefinite/definite integrals. It seems all of my homework problems have done the less complicated ones so I'm not sure how to do these ones:

First problem: INDEFINITE
(integral symbol)((2x+3)/(x^2+3x-10)^3))dx

Second problem: DEFINITE
(integral symbol-WITH 0 AT BOTTOM AND 1 AT TOP)((e^(2x)/(1+sqrt of e^2x))dx

I'm very confused on how to do most of these so if there's a basic way to do it I'd love to learn it as I can't post every problem on here (although each one is more complicated than the last)

Thank you!

Answer 1

First problem: whenever you are presented with a fraction, use the following technique.

 

Let u = the denominator.  In this case u = x^2 + 3x - 10. This makes du = (2x + 3) dx

 

Now you can rewrite the original function as (intregral symbol) du/u.

 

This integration is equal to ln u + C where ln is the symbol for the natural logarithm

 

You may now replace u with (x^2 + 3x - 10).

 

The final result is ln (x^2 + 3x - 10) + C        (C represents any arbitrary constant.)

Answer 2

First problem: INDEFINITE
(integral symbol)((2x+3) (x^2+3x-10)^3))dx   NOTE that I erred in copying the problem (I missed the division symbol)

 

SOLUTION: Let u = x²+3x-10; then du = (2x+3)dx, so rewrite the problem as

∫ u³ du and now the antidifferentiation is easy; switch back to the x-based version afterwards (& don't forget +C).

 

Second problem: integrand is (e^(2x)/(1+sqrt of e^2x))dx.

Let u = 1 + e^x which can be substituted for (1+sqrt of e^2x). Also, du = e^x dx.

 

Rewrite the indefinite integral as ∫ [e^x/u] e^x dx = ∫ [(u-1)/u] du = ∫ du - ∫ (1/u) du

and I will leave it to you to properly antidifferentiate this, and then to apply FTC with definite limits as stated for this 2nd problem.