How do I use substitution to solve these integrals?

In summary, integration by substitution is a method used to find the integral of a function by making a substitution for a variable in the original function. It is most useful when the integrand contains a composition of functions, and the steps to perform it include identifying a suitable variable substitution, rewriting the original function, calculating the differential, substituting into the integral, integrating in terms of u, and substituting back in the original variable. Some common mistakes to avoid include forgetting to take the derivative, not substituting back into the final answer, choosing an unsuitable substitution, and forgetting to include the limits of integration. Integration by substitution is not always the best method for solving integrals and other techniques may be more suitable for different types of integrands.
  • #1
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Please help me calculate these integrals.
Suppose that ∫f(t)dt=11 [0,1]. Calculate each of the following.
A. ∫f(4t)dt [0,0.25] =

B. ∫f(1−4t)dt [0,0.25] =

C. ∫f(3−8t)dt [0.25,0.375] =

I did the first one and I believe is right and I try to do the 2nd and 3rd one similarily and can't get them .:uhh:
u=4t
du=4dt, or
dt=(1/4)du
∫f(4t)dt [0,0.25]
=∫f(u)(1/4)du [0,4*0.25]
=(1/4)∫f(u)du [0,1]
=(1/4)*11
=11/4
 
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  • #2
The second one is almost exactly the same. What substitution would you try??
 

1. What is integration by substitution?

Integration by substitution is a method used to find the integral of a function by making a substitution for a variable in the original function. This method is also known as u-substitution or change of variables.

2. When should I use integration by substitution?

Integration by substitution is most useful when the integrand (the function being integrated) contains a composition of functions, such as f(g(x)), where g(x) is the inner function. In this case, substitution allows us to rewrite the integrand in a simpler form before integrating.

3. How do I perform integration by substitution?

To perform integration by substitution, follow these steps:

1. Identify a suitable variable substitution, often denoted by u.

2. Rewrite the original function in terms of u.

3. Calculate the differential du/dx.

4. Substitute the new expression for u and du/dx into the integral.

5. Integrate the resulting expression in terms of u.

6. Substitute back in the original variable to get the final answer.

4. What are some common mistakes to avoid when using integration by substitution?

Some common mistakes to avoid when using integration by substitution include:

- Forgetting to take the derivative of the new variable when calculating du/dx.

- Not substituting the original variable back into the final answer.

- Choosing an unsuitable substitution, resulting in a more complicated integral.

- Forgetting to include the appropriate limits of integration in the final answer.

5. Can integration by substitution be used to solve all integrals?

No, integration by substitution is not always the best method for solving integrals. It is most effective when the integrand contains a composition of functions. Other integration techniques, such as integration by parts or trigonometric substitution, may be more suitable for other types of integrands.

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