Homework 2 Solutions Exercise 1. function.davissch/previous_classes/224_summer2011/224... ·...
Transcript of Homework 2 Solutions Exercise 1. function.davissch/previous_classes/224_summer2011/224... ·...
Homework 2Solutions
Exercise 1. Use the Fundamental Theorem of Calculus to find the derivative of thefunction.
1. g(x) =∫ x0
√1 + 2t dt
2. F (x) =∫ 10x tan θ dθ [Hint:
∫ 10x f(θ) dθ = −
∫ x10 f(θ) dθ]
3. g(x) =∫ 3x2x
u2−1u2+1
du
Exercise 2. Find the average value of the function on the given interval.
1. f(x) = x2, [−1, 1]
2. g(x) = cosx, [0, π/2]
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Exercise 3. Evaluate the integral by making the given substitution.
1.∫cos 3x dx, u = 3x
2.∫x2√x3 + 1 dx, u = x3 + 1
Exercise 4. Evaluate the integral using integration by parts with the indicated choicesof u and dv.
1.∫x lnx dx, u = lnx, dv = xdx
2.∫θ sec2 θ dθ, u = θ, dv = sec2 θdθ
2
Exercise 5. Decide which integration technique (substitution, int. by parts) is appro-priate and evaluate the integral.
1.∫2x(x2 + 3)4 dx
2.∫x cos 5x dx
3.∫ (lnx)2
x dx
4.∫cos θ sin6 θ dθ
5.∫ 1/20 sin−1 x dx
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