1. ## Definite integral question

Originally Posted by Chris L T521
For this to be a pdf, $\displaystyle \int_{-\infty}^{\infty}P\left(x\right)\,dx=1$

Thus, solve the equation $\displaystyle \int_{-\infty}^{\infty} k\left(1-x^3\right)\,dx=1\implies \int_0^1 k\left(1-x^3\right)\,dx=1$ for k.

Can you take it from here?
see this question below(integration)They are asking that the integral be evaluated.
is the procedure just like what we did(integration in the above question?)

$\displaystyle \int_{0}^{1}\frac{1}{x^{2}+3x+2}dx=\int_{0}^{1}\fr ac{1}{x+1}dx-\int_{0}^{1}\frac{1}{x+2}dx$

2. Originally Posted by Pringles
Originally Posted by Chris L T521
For this to be a pdf, $\displaystyle \int_{-\infty}^{\infty}P\left(x\right)\,dx=1$

Thus, solve the equation $\displaystyle \int_{-\infty}^{\infty} k\left(1-x^3\right)\,dx=1\implies \int_0^1 k\left(1-x^3\right)\,dx=1$ for k.

Can you take it from here?
see this question below(integration)They are asking that the integral be evaluated.
is the procedure just like what we did(integration in the above question?)

$\displaystyle \int_{0}^{1}\frac{1}{x^{2}+3x+2}dx=\int_{0}^{1}\fr ac{1}{x+1}dx-\int_{0}^{1}\frac{1}{x+2}dx$
It has nothing to do with that.

You're expected to know and apply the standard form $\displaystyle \int \frac{1}{x + a} \, dx = \ln |x + a|$ (I have omitted the arbitrary constant).

3. Originally Posted by mr fantastic
It has nothing to do with that.

You're expected to know and apply the standard form $\displaystyle \int \frac{1}{x + a} \, dx = \ln |x + a|$ (I have omitted the arbitrary constant).
No disrespect meant, but I am wondering why you don't just include the arbitrary constant in your posts about integration? Surely it's easier and more mathematically rigorous to write "+ C" rather than "(I have omitted the arbitrary constant)".

4. Originally Posted by Pringles
Evaluate the integral

$\displaystyle \int_{0}^{1}\frac{1}{x^{2}+3x+2}dx=\int_{0}^{1}\fr ac{1}{x+1}dx-\int_{0}^{1}\frac{1}{x+2}dx$

I would appreciate it if someone would be kind enough to give me the full workings with some explanation.

Thankyou
Do you know how expression was split into two?

$\displaystyle \displaystyle \int_{0}^{1}\frac{1}{x^{2}+3x+2}dx=\int_{0}^{1}\fr ac{1}{x+1}dx-\int_{0}^{1}\frac{1}{x+2}dx$

The general rule for such integrals is $\displaystyle \displaystyle \int \frac{1}{x+a}dx = \ln{|x+a|} + C$. Hence:

$\displaystyle \displaystyle \int_{0}^{1}\frac{1}{x+1}dx-\int_{0}^{1}\frac{1}{x+2}dx = [\ln{|x+1|}]^1_0 -[\ln{|x+2|}]^1_0$

$\displaystyle \displaystyle = [\ln{|1+1|}-\ln{|0+1|}]-[\ln{|1+2|}-\ln{|0+2|}]$

$\displaystyle =\displaystyle \ln{|2|}-\ln{|1|}-\ln{|3|}+\ln{|2|}$

$\displaystyle =\displaystyle 2\ln{|2|}-\ln{|3|}$ since $\displaystyle \displaystyle \ln{|1|} = 0$

$\displaystyle =\displaystyle \ln{|2^2|}-\ln{|3|}$ since $\displaystyle \displaystyle a\ln{|b|} = \ln{|b^a|}$

$\displaystyle =\displaystyle \ln{|4|}-\ln{|3|}$

$\displaystyle =\displaystyle \ln{|\frac{4}{3}|}$ since $\displaystyle \displaystyle \ln{|a|}-\ln{|b|}=\ln{|\frac{a}{b}|}$

5. Originally Posted by Mush
No disrespect meant, but I am wondering why you don't just include the arbitrary constant in your posts about integration? Surely it's easier and more mathematically rigorous to write "+ C" rather than "(I have omitted the arbitrary constant)".
If the result is to be used by the OP in a definite integral then I omit the constant (and say so).

If the result is part of an indefinite integral that the OP is trying to solve then I include the constant.

My reasons for doing this are not too difficult to fathom I think ....