1) Integrate the following (partial fractions). a) ? 1/(1-x²) dx b) ? 1/((1-x)²(2-x)) dx c) ? (2x-1)/(1-x?) dx d) ? 1/(1-x³) dx 12) Evaluate the following improper integrals. a) ??? 1/??x? dx b) ??? ln(x)/x dx c) ??¹ 1/(1-x²) dx d) Does ??? 1/(1+x?) dx converge? e) Does ??? e??²/x dx converge? f) Does ??² 1/(1-x³) dx converge? 13) a) Find the area of the circle x² + (y-2)² = 4 using polar coordinates. b) Find the area of the region inside the cardioid r = 1 + cos(?) and above the line y = x. c) Find the area of the region inside one leaf of the three-leaved rose r = cos(3?). d) Find the length of the spiral r = ? from 0 ? ? ? 2?.
Added by Elisa F.
Close
Step 1
We can write: J (x+1)/(x^2-x-2) dx = J (x+1)/[(x-2)(x+1)] dx = J [A/(x-2) + B/(x+1)] dx Solving for A and B, we get A = 1/3 and B = -1/3. Show more…
Show all steps
Your feedback will help us improve your experience
Adi S and 82 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
The integral ∫ x² e²ˣ dx can be evaluated using OA. Integration by parts once. OB. Integration by parts twice. OC. u-substitution OD. Impossible to integrate (no analytical solution). 2. Which set up is most useful for evaluating the integral ∫ sin²(3x) cos(3x) dx OA. ∫ sin²(3x) cos(3x) dx and u = cos(3x) OC. ∫ [1 - cos²(3x)] cos(3x) dx and u = cos(3x) OB. ∫ sin²(3x) cos(3x) dx and u = sin(3x) OD. ∫ [1 - cos²(3x)] cos(3x) dx and u = cos x 3. The appropriate change of variable for the integral ∫ 1 / √(x² + 25) dx is OA. x = 25 tan θ OB. x = 5 sin θ OC. x = 25 sin θ OD. x = 5 tan θ 4. The correct form of the partial fraction decomposition for 1 / (x⁴ - 1) is OA. (Ax + B)/(x² - 1) + (Cx + D)/(x² + 1) OB. A/(x - 1) + B/(x + 1) + C/(x + 1)² OC. A/(x - 1) + B/(x + 1) + C/(x² + 1) OD. A/(x - 1) + B/(x + 1) + (Cx + D)/(x² + 1) 5. The correct form of the partial fraction decomposition for 1 / ((x - 1)²(x² + 4)) is
Kumareshwaran R.
Partial credit will be awarded for this group. Your answers must be justified to receive full credit. 6. (14 points) SET UP an integral to compute the volume of the solid obtained by rotating the region bounded by the curves y = x^2 + 3 and y = 11 - x^2 about the x-axis. DO NOT EVALUATE. 7. (15 points). Consider the function f(x) = 1/((x + 2)(x^2 + 1)). (a) Write f(x) into its partial fraction decomposition. (b) Compute ∫ f(x)dx. 8. (30 points)Evaluate the following integrals (a) ∫ ln x / √ x dx. (b) ∫ dx / √(9x^2 - 4) (c) ∫[0 to ̀̑/2] cos^5 x sin^2 x dx. 9. (12 points) Consider the integral ∫[0 to 1] (x^5 + 3x^2)dx. Using the error bound for Simpson's method, find a bound for the error in terms of N, where N is the number of subdivisions of the interval [0, 1]. 10. (14 points) Using the comparison test for improper integrals, determine whether the integral ∫[0 to 1] x / √(x^5 - 1) dx converges or diverges. Justify your answer.
Sri K.
1) Evaluate the following integrals by Trigonometric Substitution. a) ∫ dx / √(9-x²) 12 pts b) ∫ 1 / (x²√(x²+4)) dx 12 pts 2) Evaluate the following Trigonometric Integrals. a) ∫ cos⁴x dx 12 pts b) ∫ tan³x sec⁵x dx 12 pts c) ∫ sin⁵x cos³x dx 12 pts 3) Show the partial fraction decomposition for the following rational function. Do not integrate. r(x) = (3x+2) / (x(x+2)(x²-x-1)(x²+3)³) 7 pts
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD