Integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$: A Solution

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In summary, the homework equation is \int t^{3} \cdot (3+t^{2}) dt. The attempt at a solution was to integrate by parts and then use the substitution u=3+t^2. However, the only way to solve the problem is to do integration by parts numerous times.
  • #1
Panphobia
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Homework Statement



[itex]\int \frac{t^{3}}{\sqrt{3 + t^{2}}}[/itex]

Homework Equations



∫udv = uv - ∫vdu

The Attempt at a Solution


So I tried integration by parts, then I had to integrate the last term with the same method, and then I got a u substitution integral, in the end I got.

[itex]\int \frac{t^{3}}{\sqrt{3 + t^{2}}}[/itex] = [itex]\frac{t^{4}}{\sqrt{3+t^{2}}\cdot 4} - \frac{t^{3}}{\sqrt{3+t^{2}}\cdot 2} - \sqrt[3/2]{3+t^{2}} +9\cdot \sqrt{3+t^{2}}[/itex]

This seems a little long for the space given for the question, so could someone confirm the correctness of this integral? Also how would I go about being more efficient in solving my integrals?
 
Last edited:
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  • #2
Are you sure this integral is copied correctly? The answer isn't pretty and is longer than your answer. In the future, try to include all of your intermediate steps as well.

Are you sure one of the ##t^3## isn't supposed to be a ##t^2##?
 
  • #3
Panphobia said:
This seems a little long for the space given for the question, so could someone confirm the correctness of this integral?
Don't let the space provided in a question throw you off. It has no value when solving problems. Don't get caught up in head games.

If you rewrite the question as [itex]\int t^3 \cdot (3+t^3)^\frac{-1}{2} dt[/itex], it might be a bit easier.

The only way I can see this problem being solved is integration by parts numerous times to eliminate the t3 term.
It's very painful but is good practice.
 
  • #4
scurty said:
Are you sure this integral is copied correctly? The answer isn't pretty and is longer than your answer. In the future, try to include all of your intermediate steps as well.

Are you sure one of the ##t^3## isn't supposed to be a ##t^2##?

I corrected it, my question was wrong.
 
  • #5
Panphobia said:
I corrected it, my question was wrong.
That makes a huge difference.

Show how you do the first integration by parts.

Added in Edit:

See Dick's reply (next). That should work fine.
 
Last edited:
  • #6
Panphobia said:
I corrected it, my question was wrong.

Alright, then just start with the substitution u=3+t^2. I don't think you need integration by parts at all. Show your work.
 

FAQ: Integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$: A Solution

What is the purpose of integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$?

The purpose of integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$ is to find the area under the curve of the given function. This is an important concept in calculus and is used to solve various problems in physics, engineering, and other scientific fields.

2. How do you solve the integral of $\frac{t^{3}}{\sqrt{3 + t^{2}}}$?

To solve the integral of $\frac{t^{3}}{\sqrt{3 + t^{2}}}$, we can use the substitution method. Let $u = 3 + t^{2}$, then $du = 2t\,dt$. Substituting this into the integral, we get $\int \frac{t^{3}}{\sqrt{3 + t^{2}}} \,dt = \frac{1}{2} \int \frac{t^{2}}{\sqrt{u}}\,du$. From here, we can use basic integration techniques to solve for the final answer.

3. What are the applications of integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$?

The applications of integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$ include finding the displacement, velocity, and acceleration of an object moving along a curved path, calculating the work done by a varying force, and determining the center of mass of a non-uniform object.

4. Can we use any other method to solve the integral of $\frac{t^{3}}{\sqrt{3 + t^{2}}}$?

Yes, there are other methods that can be used to solve the integral of $\frac{t^{3}}{\sqrt{3 + t^{2}}}$, such as integration by parts and trigonometric substitutions. However, in this case, the substitution method is the most efficient and straightforward approach.

5. Are there any real-life situations where integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$ is useful?

Yes, there are many real-life situations where integrating $\frac{t^{3}}{\sqrt{3 + t^{2}}}$ is useful. Some examples include calculating the amount of work done by a varying force in lifting an object, determining the path of a projectile under the influence of gravity, and finding the trajectory of a satellite in orbit around a planet.

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