How Can Integration by Parts Simplify Calculus Challenges?

[beaf123]

Hi all. I am having Calculus 1 this year. We are using a book called Thomas Calculus.
I think its a lot of fun, but I have to work very much since there is basic stuff like trigonometry that I know really bad. Since I work so much with math I thought it could be fun and helpful to talk with other math people in here:-)

To the question:

∫ (5x)/(3x^2+5) dx

What I did was this but I think its too complicated:

∫ (1/(3x^2+5)) * 5x

Integration by parts give.

(5x) ln(3x^2+5) - 5 ∫ ln (3x^2+5)dx

Not sure have to calculate the last integral. Not sure about anything here really..

 

[hedipaldi]

compute by substitution:u=3x^2+5

 

[bradcito]

A new question: With u=3x^2+5 the answer is 5/6*log(3*x^2+5), buit using symbolics in MATLAB the answer is 5/6*log(x^2+5/3) and that's using u=x^2+5/3. Why are there different answers?
As 5/6*log(3*x^2+5) = 5/6*log(x^2+5/3)/log(3) <> 5/6*log(x^2+5/3)

 

[SammyS]

A new question: With u=3x^2+5 the answer is 5/6*log(3*x^2+5), buit using symbolics in MATLAB the answer is 5/6*log(x^2+5/3) and that's using u=x^2+5/3. Why are there different answers?
As 5/6*log(3*x^2+5) = 5/6*log(x^2+5/3)/log(3) <> 5/6*log(x^2+5/3)

The two answers differ only by a constant. Remember the constant of integration?

\displaystyle \frac{5}{6}\log(3x^2+5)=\frac{5}{6}\log\left(3 \left(x^2+\frac{5}{3}\right)\right)

\displaystyle =\frac{5}{6}\log(3)+\frac{5}{6}\log\left(x^2+\frac{5}{3}\right)​

 

[beaf123]

Yes of course. I should have thought of that.
Would you get the same answer using integration by parts?

If the exercise looked like this instead:

∫ (5x)/(3x^2+4x+5) dx

then you have to use integration by parts?

 

[tiny-tim]

welcome to pf!

hi beaf123! welcome to pf!

(try using the X² button just above the Reply box )

Would you get the same answer using integration by parts?

integrating by parts, your u would be the whole thing, and your v would be 1

(your line starting "(5x) ln(3x^2+5) …" was wrong)

If the exercise looked like this instead:

∫ (5x)/(3x^2+4x+5) dx

then you have to use integration by parts?

no, write the integrand A(6x+4)/(3x²+4x+5) + B/(3x²+4x+5), and do two different substitutions

 

[beaf123]

Thank you:)