Math topic中文

By parts

Integration by parts.

Practice questions

  1. 15 (a) Solve the following integral. ∫1−2x3+2x−x2 dx.\displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}}\,\mathrm{d}x. ∫xln⁡(2−x2) dx\displaystyle\int x\ln\left(2-x^2\right)\,\mathrm{d}x…
  2. 2Find ∫3ex5−0.3ex dx\displaystyle \int \frac{3e^x}{5 - 0.3e^x} \, dx ∫cos⁡(ln⁡x) dx\displaystyle \int \cos(\ln x) \, dx, where x>0x > 0 the exact…
  3. 3Show that for x>1x>1, ddx(1x2−1)=−x(x2−1)3/2 .\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=-\dfrac{x}{(x^2-1)^{3/2}}\,.…
  4. 4Show that ∫nn+1ln⁡x dx=(n+1)ln⁡(n+1)−nln⁡n−1.\displaystyle\int_{n}^{n+1} \ln x\, dx = (n+1)\ln(n+1) - n\ln n - 1. (ii) The diagram shows the graph of y=ln⁡xy = \ln x…
  5. 5You are given that C=∫0πe−2xcos⁡x  dxC = \int_{0}^{\pi} \mathrm{e}^{-2x} \cos x \; \mathrm{d}x and S=∫0πe−2xsin⁡x  dxS = \int_{0}^{\pi} \mathrm{e}^{-2x} \sin x \; \mathrm{d}x…
  6. 6A function f is defined as f(x)=ln⁡(5−x),  x<5.f(x)=\ln(5-x),\; x<5. Find the volume of revolution when the region bounded by the curve of…
  7. 7Find ∫x2e−2x dx\displaystyle\int x^{2}e^{-2x}\,\mathrm{d}x. The curve with equation y=xe−xy = xe^{-x} and the line with equation y=1exy=\tfrac{1}{e}x…
  8. 8The curve CC has equation y=ln⁡(2−∣x∣)y=\ln\left(2-\lvert x\rvert\right). The region RR is bounded by CC, the lines y=ln⁡2y=\ln 2,…
  9. 9A farmer has a plot of the land, OABOAB, with adjacent sides OAOA and OBOB. With reference to OO as the origin, OAOA and…
  10. 10Find ∫sin⁡3xcos⁡x dx\displaystyle \int \sin 3x \cos x\, dx. Find ∫e2xcos⁡3x dx\displaystyle \int e^{2x} \cos 3x\, dx.
  11. 11Find ∫xln⁡(x2−4)dx\int x \ln(x^2 - 4) \text{d}x. Use the substitution x=acos⁡ux = a \cos u to find ∫0aa2−x2 dx\int_{0}^{a} \sqrt{a^2 - x^2} \text{ d}x…
  12. 12Find ∫x2e−2x dx\int x^{2} e^{-2x}\, dx . The curve with equation y=xe−xy = x e^{-x} and the line with equation y=1exy = \tfrac{1}{e} x…
  13. 13[Cambridge/N78/I/20] Find ddx(tan⁡−1(x2))\frac{\text{d}}{\text{d}x}(\tan^{-1}(x^2)) and hence, or otherwise, find ∫xtan⁡−1(x2) dx\int x \tan^{-1}(x^2) \, dx…
  14. 14The curve CC has equation y=xcos⁡2xy = x\cos 2x, where 0≤x≤π0 \le x \le \pi. Find the exact xx-coordinates of the points where CC…
  15. 15Find ∫sin⁡pxcos⁡qx dx\int \sin px \cos qx \text{ d}x, where pp and qq are constants such that p≠qp \ne q and p≠−qp \ne -q. Given that m≠0m \ne 0…
  16. 16Find ∫x+2x2+2x−3 dx\int \frac{x+2}{x^2+2x-3} \,\mathrm{d}x. Find ∫2xsin⁡x dx\int 2x\sin x \,\mathrm{d}x. Using the substitution x=3sin⁡θx = 3\sin\theta…
  17. 17Find ∫6+2x1−4x−x2 dx\displaystyle \int \dfrac{6+2x}{\sqrt{1-4x-x^2}}\,\mathrm{d}x. The diagram above shows the region RR bounded by…
  18. 18Find ∫xsin⁡2x dx\displaystyle \int x \sin^{2} x \, dx . The region R is bounded by the curve y=xsin⁡xy = \sqrt{x \sin x}, the lines x =…
  19. 19Integrate by parts twice to show that ∫e2xsin⁡x dx=15e2x(2sin⁡x−cos⁡x)+c.\int e^{2x}\sin x\,dx=\tfrac{1}{5}e^{2x}(2\sin x-\cos x)+c. The diagram below…
  20. 20Find ∫1−3x1+9x2 dx\int \frac{1-3x}{1+9x^2} \, \text{d}x. Use the substitution x=2sin⁡θx = 2\sin\theta, where θ\theta is acute, to find ∫(x−1)24−x2 dx\int \frac{(x-1)^2}{\sqrt{4-x^2}} \, \text{d}x…
  21. 21The diagram below shows a region RR bounded by the curve (y+5)2=x−3(y+5)^{2}=x-3 and the line y=x−10y=x-10. Find the volume of soli…
  22. 22Find ddx(esin⁡−1x1−x2)\frac{\mathrm{d}}{\mathrm{d}x} \left( e^{\sin^{-1} x} \sqrt{1 - x^2} \right). Hence using integration by parts, f…
  23. 23[Cambridge/9709/2007] Use integration by parts to show that ∫24ln⁡x dx=6ln⁡2−2.\int_{2}^{4} \ln x \, dx = 6 \ln 2 - 2.
  24. 24Use the substitution t=ext = e^{x} to show that ∫e2x tan⁡−1(ex) dx=∫t tan⁡−1(t) dt.\int e^{2x}\,\tan^{-1}\bigl(e^{x}\bigr)\,dx = \int t\,\tan^{-1}(t)\,dt.…
  25. 25Find ∫19−4x2 dx\int \frac{1}{9 - 4x^2} \text{ d}x. Using the substitution x=cos⁡θ\sqrt{x} = \cos\theta, find ∫x1−x dx\int \sqrt{\frac{x}{1-x}} \text{ d}x…
  26. 26Find ∫10ex5−2ex dx ,\int \frac{10\mathrm{e}^x}{5 - 2\mathrm{e}^x} \,\mathrm{d}x \text{ ,} ∫x1+8x2 dx ,\int \frac{x}{\sqrt{1 + 8x^2}} \,\mathrm{d}x \text{ ,}…
  27. 27Find ddxe−x2\dfrac{d}{dx}e^{-x^{2}}. Hence find ∫x3e−x2 dx\displaystyle \int x^{3}e^{-x^{2}}\,dx. By using the substitution z=ex2yz = e^{x^{2}}y…
  28. 28Find the derivative of ex2+2x\text{e}^{x^2+2x}. Hence, find ∫(x+1)3ex2+2x dx\int (x+1)^3 \text{e}^{x^2+2x} \text{ d}x.
  29. 29Differentiate x2−1\sqrt{x^2 - 1} with respect to xx. Hence find ∫xcos⁡−1(1x) dx\displaystyle \int x\cos^{-1}\left(\dfrac{1}{x}\right)\,dx…
  30. 30Find the following integrals. ∫cos⁡3x−cosec23xsin⁡3x+cot⁡3x dx\int \frac{\cos 3x - \text{cosec}^2 3x}{\sin 3x + \cot 3x} \text{ d}x ∫1−x1−16x2 dx\int \frac{1-x}{\sqrt{1-16x^2}} \text{ d}x…
  31. 31Using integration by parts, or otherwise, find ∫1yln⁡y dy\displaystyle \int \frac{1}{y}\ln y\,dy. Given that y>1y>1 and dydx=yln⁡y\displaystyle \frac{dy}{dx}=\frac{y}{\ln y}…
  32. 32A curve C has parametric equations x=et,  y=sin⁡tx = e^{t},\ \ y = \sin t where 0≤t≤π0 \le t \le \pi. Find ∫etsin⁡t dt\int e^{t}\sin t\,dt . Ske…
  33. 33Find ∫tan⁡2(x−1) dx,\int \tan^2(x - 1) \text{ d}x , ∫sin⁡−12x dx.\int \sin^{-1} 2x \text{ d}x .
  34. 34A curve P is defined by y=−sin⁡2xy = -\sin^2 x where −π≤x≤0-\pi \le x \le 0. The finite region R is bounded by the curve P, the y-a…
  35. 35Find ∫cos⁡xcos⁡3x+cos⁡x dx\int \frac{\cos x}{\cos 3x + \cos x} \text{ d}x. Find ∫xtan⁡−1x2 dx\int x \tan^{-1} x^2 \text{ d}x. Hence find the exact val…
  36. 36[In this question, all measurements of length are in metres.] A monument is to be constructed which comprises two parts…
  37. 37State the constants AA and BB such that, for all values of xx, 11−4x=A(4−2x)+B.11 - 4x = A(4 - 2x) + B. Hence, find ∫11−4xx(4−x) dx\int \frac{11 - 4x}{\sqrt{x(4 - x)}} \, \mathrm{d}x…
  38. 38The curve CC has equation y=ln⁡(2−∣x∣)y=\ln\left(2-\lvert x\rvert\right). The region RR is bounded by CC, the lines y=ln⁡2y=\ln 2,…
  39. 39[Cambridge/9233/2005] State the derivative of sin⁡(x2)\sin(x^2). Find ∫x3cos⁡(x2) dx\int x^3 \cos(x^2) \, dx.
  40. 40Using standard series from the List of Formula (MF26), expand ln⁡(k+2x)n\ln(k + 2x)^n in ascending powers of xx, as far as the…
  41. 41The region RR is bounded by the curve with equation y=xln⁡xy = x \ln x, the lines x=ex = \text{e}, x=e2x = \text{e}^2 and the xx…
  42. 42Find ∫cos⁡(ln⁡x)x2 dx\int \frac{\cos(\ln x)}{x^2} \, \mathrm{d}x. Using the substitution u=x+3u = \sqrt{x+3}, find ∫16x−2xx+3 dx\int_1^6 \frac{x-2}{x\sqrt{x+3}} \, \mathrm{d}x…
  43. 43Find ∫xx2+4x+7 dx\int \frac{x}{x^2 + 4x + 7} \, \mathrm{d}x. Find ∫0axa−x dx\int_0^a x\sqrt{a-x} \, \mathrm{d}x by using integration by pa…
  44. 44Find the general solution of the differential equation cosec x d2ydx2=x\frac{d^{2}y}{dx^{2}} = x . Find the general solution o…
  45. 45Section A: Pure Mathematics [40 marks] State the derivative of tan⁡(x3)\tan(x^3). Find ∫6x5sec⁡2(x3) dx\int 6x^5 \sec^2(x^3) \, dx.
  46. 46Find ddx(esin⁡−1x1−x2).\dfrac{d}{dx}\left( e^{\sin^{-1}x}\sqrt{1-x^{2}}\right). (ii) Hence using integration by parts, find ∫xesin⁡−1x1−x2 dx\displaystyle \int x \dfrac{e^{\sin^{-1}x}}{\sqrt{1-x^{2}}}\, dx…
  47. 47Differentiate etan⁡x\text{e}^{\tan x} with respect to xx. Hence find ∫etan⁡xsec⁡4x dx\int \text{e}^{\tan x} \sec^4 x \text{ d}x.
  48. 48The curve CC (as shown in the diagram below) has equation y=x2sin⁡xy = x^2 \sin x, −π≤x≤π-\pi \le x \le \pi. Calculate the exact a…
  49. 49Use the substitution t=ext = e^{x} to show that ∫e2xtan⁡−1(ex) dx=∫ttan⁡−1(t) dt\int e^{2x}\tan^{-1}\left(e^{x}\right)\,dx = \int t\tan^{-1}(t)\,dt. Hen…