Integrate 3t^2(1+t^3)^4

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In summary, to integrate the given expression, one can use the rule \int {\left( {a + bx} \right)^n \,\,dx = \frac{{\left( {a + bx} \right)^{n + 1} }}{{a\left( {n + 1} \right)}} + c}, or the rule \int {x^n \,\,dx = \frac{{x^{n + 1} }}{{n + 1}} + c}. Alternatively, one can use substitution by looking at the derivative of the term (1 + t^3) and solving the problem easily. Another method could be expanding the factorized expression and integrating term by term.
  • #1
unique_pavadrin
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Homework Statement


Integrate the following:
[tex]
\int {3t^2 \left( {1 + t^3 } \right)^4 \,\,dt}
[/tex]


Homework Equations


[tex]
\begin{array}{l}
\int {\left( {a + bx} \right)^n \,\,dx = \frac{{\left( {a + bx} \right)^{n + 1} }}{{a\left( {n + 1} \right)}} + c} \\
\int {x^n \,\,dx = \frac{{x^{n + 1} }}{{n + 1}} + c} \\
\end{array}
[/tex]


The Attempt at a Solution


I am unsure on how to integrate problems such as these. Is there another rule? or is it a combination of rules? Many thanks to all help provided,
unique_pavadrin
 
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  • #2
One way would be to integrate by parts a few times. There's probably a quicker way that someone else may spot though.
 
  • #3
how would i integrate by parts in there situations? thanks\
 
  • #4
No need for integration by parts. Look at the "1 + t^3" term. What is it's derivitive. This problem can be solve by a simple substitution. Do you see it?
 
  • #5
TheoMcCloskey said:
No need for integration by parts. Look at the "1 + t^3" term. What is it's derivitive. This problem can be solve by a simple substitution. Do you see it?

Haha, nice.. I knew there would be a quicker way!
 
  • #6
okay i can solve the problem now, thanks all
 
  • #7
Longer than a substitution, but shorter than integration by parts a few times, would be the expansion of the factorized expression and integrate term by term.
 

What is the integral of 3t^2(1+t^3)^4?

The integral of 3t^2(1+t^3)^4 is (1/4)(1+t^3)^5 + C, where C is a constant of integration.

How do you solve the integral 3t^2(1+t^3)^4?

To solve this integral, you can use the substitution method. Let u = 1+t^3, then du = 3t^2dt. This transforms the integral to (1/3)u^4du which can be easily integrated. Once you have the solution in terms of u, substitute back in for t.

Can the integral of 3t^2(1+t^3)^4 be evaluated by parts?

Yes, the integral can be evaluated using integration by parts. Let u = 3t^2 and dv = (1+t^3)^4dt, then du = 6tdt and v = (1/4)(1+t^3)^5. This will result in the same solution as using substitution.

What is the general strategy for solving integrals?

The general strategy for solving integrals is to first identify the type of integral (i.e. polynomial, trigonometric, exponential, etc.), and then use appropriate integration techniques such as substitution, integration by parts, or trigonometric identities. It is also important to simplify the integral and use properties of integrals, such as linearity and the constant multiple rule, to make the problem easier to solve.

Can the integral of 3t^2(1+t^3)^4 be solved using a calculator?

No, this integral cannot be solved using a calculator as it requires knowledge of integration techniques. However, some calculators have built-in integration functions that can give an approximate solution to integrals.

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