Help with integration (1) involving integration by parts etc

In summary, the original problem is to solve for the indefinite integral of (7x^3)/sqrt(4+x^2)(dx). The attempt at a solution involves factoring out the constant 7 and using trig substitution, specifically setting 1/(4+x^2) as dv and u as x^3 with du as 3x^2. However, further simplification is needed and after trying a different substitution, it is discovered that an extra u was mistakenly included in the numerator. After correcting this error, the correct solution is obtained.
  • #1
NCyellow
22
0

Homework Statement


Solve for indefinite integral of
(7x^3)/sqr(4+x^2)(dx)

Homework Equations


I just can't seem to find the right solution.


The Attempt at a Solution


First of all, we can just factor the 7 out of the integral for now since it is only a constant.
the inverse square root of (4+x^2) looks like arctan(x/2).
So I set 1/(4+x^2) up as dv, and so V would equal arctan(x/2). U is then x^3, and du is 3x^2.
However, the integrals just don't seem to simplify. Please help.
 
Physics news on Phys.org
  • #2
how about looking at trig substitution - in particular what trig identity could simplfy the denominator
 
  • #3
Substitution again. Try u=x^2+4. You have a left over x^2 in the numerator. But x^2=u-4. Try the easy stuff before you resort to the hard stuff.
 
  • #4
I did it, and ended up with the integral of (u^2-4u) over square root of u, all multiplied by the constant 7/2. After a lengthy algebra session, I ended up with a huge answer, that wasn't correct... What did I do wrong?
 
  • #5
I ended up with basically (u-4)*du/sqrt(u) forgetting the constants. What did you do? I think you have an extra u in the numerator which doesn't belong there.
 
  • #6
Ah, there we go. I forgot to take out an x for du. Thanks.
 

1. What is integration by parts?

Integration by parts is a method used in calculus to evaluate integrals that are in the form of a product. It involves breaking down the integral into two parts and using the product rule from differentiation to solve it.

2. When should I use integration by parts?

Integration by parts is typically used when the integral contains a product of two functions, and attempts to use other integration techniques, such as substitution or partial fractions, have been unsuccessful.

3. How do I choose which function to differentiate and which to integrate in integration by parts?

The general rule is to choose the function to differentiate based on the order of preference: logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential. The function to integrate should be the remaining function in the integral.

4. What is the formula for integration by parts?

The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are the two functions in the integral and du and dv are their respective differentials.

5. Can integration by parts be used for definite integrals?

Yes, integration by parts can be used for definite integrals. After applying the integration by parts formula, the definite integral can be evaluated using the fundamental theorem of calculus.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
955
  • Calculus and Beyond Homework Help
Replies
10
Views
284
  • Calculus and Beyond Homework Help
Replies
4
Views
564
  • Calculus and Beyond Homework Help
Replies
22
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
449
  • Calculus and Beyond Homework Help
Replies
15
Views
744
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
222
  • Calculus and Beyond Homework Help
Replies
1
Views
805
Replies
5
Views
627
Back
Top