How to know when to use integration by substitution vs. integration by parts?
I'll outline a procedure, but in essence it's as follows: attempt substitution initially, if it doesn't yield an answer, try parts; if that doesn't yield an answer either, try an alternative method.
If you can visualize the product as having one factor that is the derivative or "almost" the derivative of the other, that's a good suggestion for substitution. If not, you may need to make multiple substitutions. If you can visualize the product as having one factor that you can differentiate and the other that you can integrate, that's a good suggestion for parts.
Certain integrals can be assessed using either approach, while the majority won't yield results using either. (On an exam, most can be done by some method you've learned.)
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To determine whether to use integration by substitution or integration by parts, follow these guidelines:

Integration by Substitution:
 Use substitution when you have a composite function, where the derivative of an inner function can be simplified.
 Look for patterns such as ( \int f(g(x)) \cdot g'(x) , dx ), where ( g'(x) ) appears as a factor and ( f(g(x)) ) can be integrated easily.
 Substitution works well for expressions involving trigonometric, exponential, or logarithmic functions.

Integration by Parts:
 Use integration by parts when you have a product of two functions, neither of which is a straightforward derivative or antiderivative of the other.
 Look for expressions of the form ( \int u , dv ), where ( u ) and ( dv ) can be chosen appropriately.
 Typically used when one factor can be differentiated to simplify it and the other factor can be integrated easily.
 Often applied to functions involving algebraic expressions, logarithmic functions, or inverse trigonometric functions.
Ultimately, the choice between integration by substitution and integration by parts depends on recognizing the structure of the integral and selecting the method that simplifies the problem most effectively. Practice and familiarity with both techniques will aid in making this decision.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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