lab9_solns_f11

# lab9_solns_f11 - Math 116 - Lab 9 Solutions - Fall 2011. 1....

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Unformatted text preview: Math 116 - Lab 9 Solutions - Fall 2011. 1. Compute the following integrals by substitution. a) I = integraltext sin(3 x ) dx . We can use the substitution u = 3 x , which gives du = u ′ ( x ) dx = 3 dx and thus du/ 3 = dx , so I = integraltext sin( u ) / 3 du =- cos( u ) / 3 + c =- cos(3 x ) / 3 + c . b) I = integraltext exp( t + 3) dt . We can use the substitution u = t + 3, which gives du = u ′ ( t ) dt = dt , so I = integraltext exp( u ) du = exp( u ) + c = exp( t + 3) + c . c) I = integraltext (3 t 3 + 7) 4 t 2 dt . We can use the substitution u = 3 t 3 + 7, which gives du = u ′ ( t ) dt = 9 t 2 dt and thus du/ 9 = t 2 dt , so I = integraltext u 4 / 9 du = u 5 / 45 + c = (3 t 3 + 7) 5 / 45 + c . d) I = integraldisplay t 3 t 2 + 5 dt . We can use the substitution u = 3 t 2 + 5, which gives du = u ′ ( t ) dt = 6 tdt and thus du/ 6 = tdt , so I = integraltext 1 / (6 u ) du = 1 / 6 ln | u | + c = 1 / 6 ln | 3 t 2 + 5 | + c . e) I = integraldisplay 3 x 2 1 + x 6 dx . (Hint: recall the derivative of arctan( x ).) We can use the substitution u = x 3 , which gives du = u ′ ( x ) dx = 3 x 2 dx , so I = integraltext 1 / (1 + u 2 ) du = arctan( u ) + c = arctan( x 3 ) + c . f) I = integraldisplay 1 t 2 + 2 t + 5 dt . (Hint: recall the derivative of arctan( x ) and complete the square.) Upon inspection, it looks like this integral may fit the pattern of integrating the derivative of arctan( x ), if we manage to complete the square in the denominator of the integrand. Completing the square we get: I = integraldisplay 1 t 2 + 2 t + 1- 1 + 5 dt = integraldisplay 1 ( t + 1) 2 + 4 dt = 1 4 integraldisplay 1 (( t + 1) / 2) 2 + 1 dt, so we can use the substitution u = ( t +1) / 2, which gives du = u ′ ( t ) dt = 1 / 2 dt and thus 2 du = dt , so...
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## This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.

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lab9_solns_f11 - Math 116 - Lab 9 Solutions - Fall 2011. 1....

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