Calculus 2_Practice Exam Solutions

Calculus 2_Practice Exam Solutions - Sample Test Solutions...

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Sample Test Solutions Problem 1 : 0πcos2x dx Step 1. Recall the half angle formula: = + cos2a 121 cos2a = - sin2a 121 cos2a Step 2. Apply the half angle formula of cos2a to the integral = + 0πcos2x dx 0π121 cos2xdx = 0πsin2x dx 120π + 1 cos2xdx Step 3. Solve the integral 120π + 1 cos2xdx 120π + =[ ( + ] 1 cos2xdx 12 x 12sin2x π 0 Step 4. Integrate + 12x 12sin2x π = + - ( )- ( ) 0 12π 14sin2π 12 0 14sin2 0 = 0πcos2x dx 12π
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Problem 2-1 : cosxsin3x dx Step 1. Use substitution = u sin x = du cosx dx = cosxsin3x dx u3du Step 2. Solve the integral: = u3du 14u4 Step 3. Write in terms of x = cosxsin3x dx + 14sin4x C
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Problem 2-2 : tan3xsec4x dx Step 1. Recall the following trig identity =( + ) sec2x 1 tan2x Step 2. Apply the trig identity to the integral = + tan3xsec4x dx tan3x1 tan2xsec2x dx Step 3. Use substitution = u tan x = du sec2x dx + = + tan3x1 tan2xsec2x dx u31 u2du Step 4. Solve the integral + = + u31 u2du 14u4 16u6 Step 5. Write solution in terms of x = tan3xsec4x dx + + 14tan4x 16tan6x C
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Problem 3 : - x2 9 dx Step 1. Recall the trigonometric substitution and trig identity
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Calculus 2_Practice Exam Solutions - Sample Test Solutions...

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