Integration by Substitution

In summary, integration by substitution is a technique used in calculus to simplify the integration of complex functions. It involves substituting a variable in the integrand with a new variable that makes the integration easier. You should use this technique when the integrand involves a function within a function, such as composite functions or nested functions, or when it contains a function and its derivative. The general process for integration by substitution involves identifying the substitution variable, finding its derivative, and substituting both in the integrand. Some common substitution variables used are u, x, t, and θ. There are also special cases, such as using trigonometric identities, inverse trigonometric functions, or logarithmic or exponential functions as substitution variables.
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Hi, I'm new to integration and I'm trying to figure out where I went wrong on this question. I'm close to the answer, but I can't tell where I've gone wrong? Can anyone help?

Thanks.

Homework Statement



Q. Evaluate the following:

The Attempt at a Solution



Please see attachment.
 

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  • #2
It might be easier to focus on the indefinite integral, and use your substitution to get an antiderivative. After that, undo your substitution. Then, evaluate your antiderivative at the two limits of integration.
 

1. What is integration by substitution?

Integration by substitution is a technique used in calculus to simplify the integration of complex functions. It involves substituting a variable in the integrand with a new variable that makes the integration easier.

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

You should use integration by substitution when the integrand involves a function within a function, such as in the case of composite functions or nested functions. It is also useful when the integrand contains a function and its derivative.

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

The general process for integration by substitution involves three steps: identifying the substitution variable, finding the derivative of the substitution variable, and substituting the original variable and its derivative in the integrand to simplify the integration.

4. What are some common substitution variables used in integration?

Some common substitution variables used in integration include u, x, t, and θ. The choice of substitution variable depends on the form of the integrand and the complexity of the function being integrated.

5. Are there any special cases in integration by substitution?

Yes, there are a few special cases in integration by substitution. These include using trigonometric identities to simplify the integrand, using the inverse trigonometric functions as substitution variables, and using logarithmic or exponential functions as substitution variables.

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