Trigonometric Integrals and Trigonometric Substitution · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Tame trig powers and rooted expressions

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. What is the correct first step for the integral of sin cubed x dx?

    • Split off one sin x and write the rest with sin squared = 1 minus cos squared
    • Split off one cos x and substitute u = sin x at once
    • Apply the half angle formula to sin cubed directly
    • Write sin cubed as sin x times cos squared x

  2. What is the correct first step for the integral of sin^3 x dx?

    • Split off one sin x factor
    • Apply a half-angle formula at once
    • Set u = sin^3 x

  3. An integrand contains sqrt(16 - x^2). In x = a sin theta, what is a?

    • 2
    • 16
    • 4

  4. After splitting sin^3 x, which identity converts what is left?

    • sin^2 x = 1 + cos^2 x
    • sin^2 x = 1 - cos^2 x
    • sin^2 x = 2 cos^2 x

  5. What is the correct first step for the integral of cos to the fifth x dx?

    • Split off one cos x and write the rest with cos squared = 1 minus sin squared
    • Split off one sin x and write everything in cosine
    • Apply the half angle formula four times in a row
    • Write cos to the fifth as cos x times tan to the fourth

  6. Use x = sin theta to evaluate the integral from 0 to 1 of x/sqrt(1 - x^2) dx.

    Answer: ______________

  7. Which substitution fits an integrand with sqrt(x^2 + 4)?

    • x = 2 sin theta
    • x = 2 tan theta
    • x = 2 sec theta

  8. Which substitution fits an integrand containing sqrt of x squared plus 4?

    • x = 2 tan theta
    • x = 2 sin theta
    • x = 4 tan theta
    • x = 2 sec theta

  9. With x = 2 sin theta you finish with an expression in theta. How do you return to x?

    • Draw a right triangle and read off the sides
    • Replace theta with x directly
    • Differentiate the theta answer

  10. With x = 2 sin theta you finish integrating and hold an expression in theta. How do you return to x?

    • Draw a right triangle with sin theta = x over 2 and read off the other sides
    • Replace every theta with x divided by 2
    • Differentiate the theta expression with respect to x
    • Set theta to arcsin x and leave it in inverse form
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Tame trig powers and rooted expressions W1-mt_JqysS41psg-s1

  1. Split off one sin x and write the rest with sin squared = 1 minus cos squared · An odd power of sine gives up one factor as du, and the leftover even power converts to cosine.
  2. Split off one sin x factor · An odd power gives up one factor for du, leaving an even power to convert.
  3. 4 · Match sqrt(a^2 - x^2) to sqrt(16 - x^2), so a^2 = 16 and a = 4.
  4. sin^2 x = 1 - cos^2 x · The Pythagoras identity gives sin^2 x = 1 - cos^2 x, ready for u = cos x.
  5. Split off one cos x and write the rest with cos squared = 1 minus sin squared · An odd power of cosine gives up one factor as du for u equals sin x, and the rest converts via the Pythagoras identity.
  6. 1 · The integrand becomes sin theta from 0 to pi/2, which integrates to 1.
  7. x = 2 tan theta · A sum of squares under a root wants x = a tan theta, here with a = 2.
  8. x = 2 tan theta · A sum of squares under a root calls for tangent, with a squared equal to 4 so a is 2.
  9. Draw a right triangle and read off the sides · A right triangle with opposite side x and hypotenuse 2 translates every trig piece back.
  10. Draw a right triangle with sin theta = x over 2 and read off the other sides · The substitution defines a triangle with hypotenuse 2 and opposite side x, from which cos theta and tan theta follow.
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