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Techniques of Integration - Integration by Substitution

Techniques of Integration - Integration by Substitution

    1. Find the integral of \(\int\frac{x}{\sqrt{1-x^2}}\,\mathrm dx\).
    2. Hence, find the exact value of \(\int_0^{\frac12}\frac{x^3}{\sqrt{1-x^2}}\,\mathrm dx\).
  2. Find \(\int \frac{2+\cos x}{\sin^2x}\,\mathrm dx\).
  3. Find \(\int \sin x(1+\cos^2x)\,\mathrm dx\).
  4. Integrate \(\int x^2\cos2x\,\mathrm dx\).
  5. Show that \(\int_0^{\frac\pi4}x\tan^2x\,\mathrm dx=\frac\pi{32}(8-\pi)-\ln\sqrt2\)
  6. Find \(\int \frac{\sin\theta\cos\theta}{(3+5\sin^2\theta-2\cos^2\theta)^{\frac13}}\,\mathrm d\theta\).
  7. Show that \(\int_0^{\frac\pi4}(1-\tan x)^2\,\mathrm dx=1-\ln2\).
  8. \(\int \frac{2x+1}{\sqrt{x+1}}\,\mathrm dx\)
  9. Find \(\int \frac{x-7}{2x^2-3x-2}\,\mathrm dx\).
  10. Find \(\int_0^{\frac\pi3}\sin^3x\,\mathrm dx\).
  11. Evaluate \(\int_0^{\frac\pi4}\sin3x\sin x\,\mathrm dx\).
  12. Find \(\int \sin^2\theta\cos^3\theta\,\mathrm d\theta\).
  13. Find \(\int \operatorname{cosec}^2\theta\sqrt{\cot\theta}\,\mathrm d\theta\).
  14. Integrate \(\int \cos^2\theta(\sin\theta+\sec\theta\operatorname{cosec}\theta)\,\mathrm d\theta\).
  15. Find \(\int \sin^3\theta\cos^2\theta\,\mathrm d\theta\).
  16. \(\int \sqrt{x^2-5}\,\mathrm dx\)
  17. \(\int 3^{\sqrt{2x+1}}\,\mathrm dx\)
  18. \(\int \frac{x\sin^{-1}(x^2)}{\sqrt{1-x^4}}\,\mathrm dx\)

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Answer:(1)(a) \(-\sqrt{1-x^2}+C\). (1)(b) \(\frac23-\frac{3\sqrt3}{8}\). (2) \(-2\cot x-\operatorname{cosec}x+C\). (3) \(-\cos x-\frac{\cos^3x}{3}+C\). (4) \(\frac{x^2\sin2x}{2}+\frac{x\cos2x}{2}-\frac{\sin2x}{4}+C\). (5) \(\frac\pi{32}(8-\pi)-\ln\sqrt2\). (6) \(\frac3{28}(1+7\sin^2\theta)^{\frac23}+C\). (7) \(1-\ln2\). (8) \(\frac43(x+1)^{\frac32}-2\sqrt{x+1}+C\). (9) \(\frac32\ln|2x+1|-\ln|x-2|+C\). (10) \(\frac5{24}\). (11) \(\frac14\). (12) \(\frac{\sin^3\theta}{3}-\frac{\sin^5\theta}{5}+C\). (13) \(-\frac23(\cot\theta)^{\frac32}+C\). (14) \(-\frac{\cos^3\theta}{3}+\ln|\sin\theta|+C\). (15) \(-\frac{\cos^3\theta}{3}+\frac{\cos^5\theta}{5}+C\). (16) \(\frac{x\sqrt{x^2-5}}2-\frac52\ln|x+\sqrt{x^2-5}|+C\), for \(x>\sqrt5\) or \(x<-\sqrt5\), with independent constants. (17) \(\frac{3^{\sqrt{2x+1}}}{\ln3}\left(\sqrt{2x+1}-\frac1{\ln3}\right)+C\). (18) \(\frac14[\sin^{-1}(x^2)]^2+C\).

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