Substitution

Substitution - THE SUBSTITUTION RULE Example 1. Find 3x2 1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
THE SUBSTITUTION RULE Example 1. Find Z 3 x 2 p 1 + x 3 dx . Solution Let u = 1+ x 3 . Then, the differential du = 3 x 2 dx . We can rewrite the indefinite integral as Z u du . Since Z u du = 2 3 u 3 / 2 + C , Z 3 x 2 p 1 + x 3 dx = 2 3 (1 + x 3 ) 3 / 2 + C The technique we used here is called integration by substitution. In general, if we have an integral of the form Z F 0 ( g ( x )) g 0 ( x ) dx, then, by the Chain rule, d dx [ F ( g ( x )] = F 0 ( g ( x )) g 0 ( x ) and Z F 0 ( g ( x )) g 0 ( x ) dx = F ( g ( x )) + C. To make it clear, we usually make the change of variable or substitution u = g ( x ) and get the following rule. The Substitution Rule If u = g ( x ) is a differentiable function whose range is an interval I and f is continuous on I , then Z f ( g ( x )) g 0 ( x ) dx = Z f ( u ) du Example 2. Find Z cos(4 x ) dx . Solution If we let u = 4 x , then du = 4 dx . Thus, Z cos(4 x ) dx = Z cos u 1 4 du = 1 4 sin u + C = 1 4 sin(4 x ) + C Example 3. Find Z sec xdx . Solution
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

Substitution - THE SUBSTITUTION RULE Example 1. Find 3x2 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online