5.6 Sub rule for Def. Integrals

5.6 Sub rule for - 1 5.6 Substitution Rule for Definite Integrals The substitution rule can be applied to definite integrals However since we

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

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

Unformatted text preview: 1 5.6 Substitution Rule for Definite Integrals The substitution rule can be applied to definite integrals. However, since we change the variable of integration, the limits of integration must also be changed since they do not apply to our new variable of integration. Let u = g (x) be a differentiable function on [a, b] whose range is an interval I , and let f be a continuous function on I , then: b f g (x) · g (x) dx = a This proves the following theorem: Theorem Substitution Rule for Definite Integrals Let u = g (x). If g (x) is continuous on the interval [a, b] and f is continuous on the range of g (x), then b f g (x) · g (x) dx = a 1 Example 1 Evaluate 0 √ t5 + 2t (5t4 + 2) dt 2 Example 2 Evaluate: π 4 • 0 1 • 0 tan3 (θ) dθ = cos2 (θ) x2 dx = 1 + x3 1 • 1 2 1 x2 3 • 0 x2 1+ 12 x + 1 4 1 dx = + 2x + 1 dx = 3 Example 3 To do Use u-substitution to set up the following integrals (evaluation is not necessary): π 4 • − π 4 π 4 • 0 1 • 0 Example 4 cos(y ) sin(y ) + 2 dy = sin(x) dx = cos(x) √9 (1 + x) √ dx = x Consider the following integral evaluation via substitution: 3 B (2x − 3)h(x2 − 3x + 7) dx = 0 Find A and B . h(u) du A 4 Completing the Square and Integration of Rational Fractions Integrals of the form 1 dx − 2x + 3 can be evaluated using linear u-substitution. To determine which function u does the job, we will need to complete the square on the denominator. 5x2 Completing the square Convert the expression of the form ax2 + bx + c into a(x + d)2 + e 1. Separate the terms with variables from the constant term: (ax2 + bx)+ c b 2. Factor out a: a x2 + a x + c 3. Complete the square: a x2 + 2 2ba x + ⇒ a x2 + 2 2ba x + b 2a 2 + c−a b2 2a b2 2a − b2 2a 4. Rewrite in the final form a x2 + Examples Completing the square • 5x2 − 2x + 3 • 2x2 − 7x + 5 • 2x2 − 6x + 5 b 2a 2 + c−a b 2a 2 +c⇒ 5 2 1 dx 5x2 − 2x + 3 Example 1 Evaluate Example 2 Is there a substitution y (x) that gives Example 3 Evaluate 0 4 −3 2x2 1 dx − 6x + 5 2x2 1 dx = A arctan(y )? − 7x + 5 ...
View Full Document

This note was uploaded on 10/02/2011 for the course MATH 1206 taught by Professor Llhanks during the Fall '08 term at Virginia Tech.

Page1 / 5

5.6 Sub rule for - 1 5.6 Substitution Rule for Definite Integrals The substitution rule can be applied to definite integrals However since we

This preview shows document pages 1 - 5. Sign up to view the full document.

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