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5.6 Sub rule for Def. Integrals

5.6 Sub rule for Def. Integrals - 1 5.6 Substitution Rule...

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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: Z b a f ( g ( x ) ) · g 0 ( x ) dx = This proves the following theorem: Theorem Substitution Rule for Definite Integrals Let u = g ( x ). If g 0 ( x ) is continuous on the interval [ a, b ] and f is continuous on the range of g ( x ), then Z b a f ( g ( x ) ) · g 0 ( x ) dx = Example 1 Evaluate Z 1 0 t 5 + 2 t (5 t 4 + 2) dt
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2 Example 2 Evaluate: Z π 4 0 tan 3 ( θ ) cos 2 ( θ ) = Z 1 0 x 2 1 + x 3 dx = Z 1 1 2 1 x 2 ( 1 + 1 x ) 2 + 1 4 dx = Z 3 0 1 x 2 + 2 x + 1 dx =
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3 Example 3 To do Use u-substitution to set up the following integrals (evaluation is not neces- sary): Z π 4 - π 4 cos( y ) p sin( y ) + 2 dy = Z π 4 0 sin( x ) cos( x ) dx = Z 1 0 (1 + x ) 9 x dx = Example 4 Consider the following integral evaluation via substitution: Z 3 0 (2 x - 3) h ( x 2 - 3 x + 7) dx = Z B A h ( u ) du Find A and B .
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4 Completing the Square and Integration of Rational Fractions Integrals of the form
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