Integration by Parts and Trig Substitution

MATH 116 SI worksheet · Winter 2024 · Session 4

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I. Evaluate using integration by parts

  1. \[\displaystyle \int x\cos x\,dx\]
  2. \[\displaystyle \int x\arcsin x\,dx\]
  3. \[\displaystyle \int \arctan x\,dx\]
  4. \[\displaystyle \int x^n\ln x\,dx\]
  5. \[\displaystyle \int e^x\cos x\,dx\]
  6. \[\displaystyle \int x^n e^{x}\,dx\]
  7. \[\displaystyle \int (\ln x)^n\,dx\]
  8. \[\displaystyle \int \cos(\ln x)\,dx\]
  9. \[\displaystyle \int x\ln\left(1+\frac{1}{x}\right) dx\]
  10. \[\displaystyle \int \frac{\sqrt{x^2+1}\,[\ln(x^2+1)-2\ln x]}{x^4}\,dx\]
  11. \[\displaystyle \int \ln\left(\sqrt{1-x}+\sqrt{1+x}\right) dx\]

II. Evaluate using trigonometric substitution

  1. \[\displaystyle \int \frac{\sqrt{a^2-x^2}}{x^2}\,dx\]
  2. \[\displaystyle \int \frac{dx}{x^2\sqrt{1+x^2}}\]
  3. \[\displaystyle \int \frac{\sqrt{x^2-a^2}}{x}\,dx\]
  4. \[\displaystyle \int \frac{dx}{\sqrt{(a^2+x^2)^3}}\]

III. Extra problems I: a geometric substitution

The line from \((-1,0)\) through a point \((x,y)\) on the unit circle crosses the \(y\)-axis at \((0,t)\).

xy (x, y) (−1, 0) (0, t) t1yx θ α
  • a. Express \(\alpha\) in terms of \(\theta\). What geometric property justifies this relationship?
  • b. What is the slope of the line from \((-1, 0)\) to \((x, y)\) in terms of \(t\)? And in terms of \(x\) and \(y\)?
  • c. Using \(x^2+y^2=1\) and the slope from part b, express \(\sin\theta\), \(\cos\theta\) and \(\tan\alpha\) in terms of \(t\).
  • d. Using implicit differentiation, find \(d\theta\) in terms of \(t\).
  • e. Use parts c and d to evaluate \(\displaystyle \int \frac{d\theta}{\sin^2\theta+\cos\theta+2}.\)

IV. Extra problems II

  • a. Let \(P(x)\) be a polynomial with real coefficients. Prove that \(\displaystyle \int_{0}^{\infty} e^{-x}P(x)\,dx=P(0)+P'(0)+P''(0)+\cdots\)
  • b. Compute \(\displaystyle \int_{0}^{\pi/4} \tan^{2n}x\,dx\) for \(n\ge1\).

Part III follows Peter Magyar’s Geometric Trig Substitution notes.