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Inept
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Homework Statement
Use substitution to evaluate the integral
Homework Equations
∫√(cotx) csc^(2)x dx?
Substitution is typically used when the integrand contains a function within a function, or when there is a chain rule involved. This can also be identified by looking for a pattern in the integrand, such as a polynomial raised to a power or a trigonometric function multiplied by its derivative.
The first step is to identify the function that needs to be substituted, usually denoted by u. Then, find the derivative of u and use that to replace the function within the integral. After substituting, the integral can be evaluated using basic integration techniques.
No, substitution is not always necessary or applicable for evaluating integrals. Some integrals can be solved using other techniques, such as integration by parts or partial fractions. It is important to identify the integral and choose the appropriate method for evaluation.
In general, there is no specific order for substitutions. However, it is important to choose a substitution that will simplify the integral and make it easier to evaluate. Sometimes, multiple substitutions may be necessary to fully simplify the integral.
One common mistake is forgetting to substitute the differential, dx, when making the substitution. It is also important to be careful when choosing the function to substitute, as it can greatly affect the difficulty of the integral. It is always helpful to check the answer by differentiating it to ensure it is correct.