Calc1131W11T1 - Integration by parts

Sunday, 20 May 2018

7:52 PM

Untitled picture.png Machine generated alternative text: The integration by parts formula is one of the most useful in the integral calculus. Ifu and t' are both differentiable functions of of then the product rule O tells us that du t" dc If we integrate both sides with respect to x, we get du We can rearrange this to get dv u dc which is sometimes more compactly written as dc + uv — u dc dv u dc du v dc t' du. Going from the left hand side to the right hand side; we have replaced one integration by another This involves taking the integrand on the left and writing it as the product offtvo functions, where dv one is a derivative of a function we know. This is useful if the product is easier to integrate than u The integration by parts formula also works for definite integrals, where the formula is [select all that apply]: u uv dx [uv] uvdx. Untitled picture.png Machine generated alternative text: Using integration by parts, find the following indefinite integrals: 5x ii) x + Note: the Maple notation for the exponential ex is exp (x) 0 o åt+C
Untitled picture.png Machine generated alternative text: Find the definite integral 25 In x dc 4 Note: 2/9 (-234 + - • the Maple notation for Inx is In (x) • you can use the preview button to preview your answer. Untitled picture.png Machine generated alternative text: In some situations, you might have to apply integration by parts twice With this in mind, find sin(4 x) dc = -(xA2 * + + 0åt+C Note: you may need to recall that the Maple notation for the exponential ex is exp (x) , or that the Maple notation for sinx is sin (x) Untitled picture.png Machine generated alternative text: For the following, you might have to do something else first, before you get an integration by parts: dc = 2sin(sqrt(x)) - 25 1 0åt+C Note: the Maple notation for is sqrt (x) , and and the Maple notation for e is exp (x) Ink Drawings

 

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