courses › Calculus

Substitution, integration by parts, and improper integrals.

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Opens at level 44.

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Builds on: Integrals (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Substitution par 45 s not tried yet

  worked example

  What is ∫₀¹ 2x(x² + 2)³ dx? Enter a fraction or a whole number.

  Answer: 65/4

  1. Let u = x² + 2; then du = 2x dx. x = 0 gives u = 2 and x = 1 gives u = 3.
  2. The integral becomes ∫ u³ du from u = 2 to u = 3.
  3. That is (3⁴ − 2⁴)/4 = (81 − 16)/4 = 65/4.

• Integration by parts par 60 s not tried yet

  worked example

  What is ∫ 6x·cos(x) dx from x = 0 to x = π?

  Answer: -12

  1. By parts with u = x and dv = cos(x) dx: ∫ x·cos(x) dx = x·sin(x) + cos(x).
  2. At π: 0 + (−1) = −1. At 0: 0 + 1 = 1.
  3. So the integral is 6 × (−1 − 1) = −12.

• Improper integrals par 40 s not tried yet

  worked example

  What is ∫₁^∞ 9/x³ dx? Enter a fraction or a whole number.

  Answer: 9/2

  1. 9/x³ = 9x⁻³, so an antiderivative is 9x⁻²/(−2) = −9/(2x²).
  2. As x → ∞ it goes to 0; at x = 1 it is −9/2.
  3. So the integral is 0 − (−9/2) = 9/2.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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