Integration question

**Paymemoney** · December 20th 2009, 09:25 PM

Hi
Can someone tell me how would i anti-differentiate the following:
1) $\frac{x}{x+1}$
2) In this question you have to evaluate the following definite integrals:
This is what i done:
$\int_1^5e+\frac{1}{x}dx$
$=e^x+ln(x)$
$=e^5+ln^5-e^1+ln(1)$
$=ln(5)+e^4$

answer was ln(5) +4e but i don't understand how they got it.

P.S

**Chris L T521** · December 20th 2009, 09:42 PM

Originally Posted by Paymemoney

Hi
Can someone tell me how would i anti-differentiate the following:
1) $\frac{x}{x+1}$

Hint: $\frac{x}{x+1}=1-\frac{1}{x+1}$

2) In this question you have to evaluate the following definite integrals:
This is what i done:
$\int_1^5e+\frac{1}{x}dx$
$=e^x+ln(x)$
$=e^5+ln^5-e^1+ln(1)$
$=ln(5)+e^4$

answer was ln(5) +4e but i don't understand how they got it.

P.S

$\int e\,dx\neq e^x$ . Here, $e$ is a constant; this means we apply the rule $\int a\,dx=ax+C$ .

Also, $e^5-e^1\neq e^4$ . You can't subtract the exponents like that.

**Paymemoney** · December 20th 2009, 09:51 PM

ok i understand why it is 4e now thanks.

**Paymemoney** · December 20th 2009, 11:45 PM

Just be make sure i am right, for 1) do you divide x by x+1? or divide x+1 by x?

**oblixps** · December 21st 2009, 09:32 AM

Originally Posted by Paymemoney

Just be make sure i am right, for 1) do you divide x by x+1? or divide x+1 by x?

you divide x by x+1. use polynomial long division. or since these are simple and easy polynomials, you can use a little trick. what i do is add and subtract a 1 from the top so you have (x + 1 - 1) / (x + 1), then i rearrange: (x+1)/(x+1) - (1/(x+1)) so now the integrand becomes 1 - (1/(x+1))

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