Integration by substitution indefinite integral 5

Can you please clarify?In summary, the conversation is about finding the indefinite integral of 5\picos\pit using the substitution method. The suggested substitution is u = \pi t, and the resulting integral is 5 \pi sin(u)du. The steps taken to solve the integral are not clear and it is suggested to include an integral sign in the expression.
  • #1
char808
27
0

Homework Statement



indefinite integral 5[tex]\pi[/tex]cos[tex]\pi[/tex]t

Homework Equations


The Attempt at a Solution



5[tex]\pi[/tex] int cos[tex]\pi[/tex]t

Substitution Method

5[tex]\pi[/tex] x sin (1/[tex]\pi[/tex]t
 
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  • #2


char808 said:

Homework Statement



indefinite integral 5[tex]\pi[/tex]cos[tex]\pi[/tex]t

Homework Equations





The Attempt at a Solution



5[tex]\pi[/tex] int cos[tex]\pi[/tex]t

Substitution Method

5[tex]\pi[/tex] x sin (1/[tex]\pi[/tex]t
Is this your integral?
[tex]\int 5 \pi cos(\pi t)dt[/tex]
Click the integral above to see how the LaTeX looks.

A reasonable substitution would be u = [itex]\pi t[/itex], so du = [itex]\pi dt[/itex].
Can you take it from there?
 
  • #3


[tex]
5 \pi sin(u)du

[/tex]

[tex]

5 \pi sin(u) x du/ \pi

[/tex]

[tex]

5sin(\pi t)

[/tex]
 
  • #4


What is the first line supposed to mean?
How did you get from the first line to the second?
How did you get from the second line to the third?
Are any of these expressions related to each other in any way?
Since this is an integration problem, one would think there should be an integral sign somewhere.
 

1. What is integration by substitution?

Integration by substitution is a technique used in calculus to find the indefinite integral of a function. It involves replacing a variable in the integrand with a new variable, and then using the chain rule to find the integral.

2. How do you know when to use integration by substitution?

Integration by substitution is typically used when the integrand contains a composition of functions, such as f(g(x)). In these cases, substitution can help simplify the integral and make it easier to solve.

3. What is the general process for integration by substitution?

The general process for integration by substitution involves four steps: 1) Identify a composition of functions in the integrand, 2) Choose a new variable to replace the inner function, 3) Calculate the derivative of the new variable, and 4) Rewrite the integral in terms of the new variable and solve.

4. Can any function be solved using integration by substitution?

In theory, any function can be solved using integration by substitution. However, the effectiveness of this technique depends on the complexity of the integrand and the availability of an appropriate substitution. In some cases, other integration techniques may be more efficient.

5. Are there any common mistakes to avoid when using integration by substitution?

One common mistake when using integration by substitution is forgetting to include the derivative of the new variable in the integral. This can lead to incorrect solutions. It is also important to choose a suitable substitution that simplifies the integral, rather than making it more complicated.

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