← Calculus II

Unit 2: Integration by Substitution

The chain rule run backward — spotting the inner function whose derivative is already sitting in the integral.

5 topics · 22 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

The Idea Behind u-Substitution

  1. How it worksHow does setting u equal to the inner function turn the chain rule into a simple integral in u?
  2. Worked exampleFind the integral of 2x times cos(x2) dx using u-substitution.
  3. Worked exampleFind the integral of 3x2 times sin(x3) dx using u-substitution.
  4. Common mistakeWhy doesn't every integral with a composite function inside it fall to plain substitution?
  5. Going furtherWhat happens when the derivative of the inner function is only off by a constant factor?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 6x cos(x2), satisfying F(0) = 0.

Substitution in a Definite Integral

  1. How it worksHow does converting the bounds from x-values to u-values save you from switching back to x at the end?
  2. Worked exampleFind the integral from 0 to 2 of x times √(x2 + 1) dx using substitution.
  3. Common mistakeWhy isn't it fine to plug u back into the original x-bounds after integrating?
  4. Ties togetherHow does changing the bounds during substitution let you skip converting the antiderivative back to x?
Your turn1 of 2
Find the definite integral of 3x √(x2 + 5) dx from 0 to 2.

Integrals Involving Exponential Functions

  1. How it worksHow does substitution turn an integral like x times ex^2 into a clean exponential antiderivative?
  2. Common mistakeWhy doesn't the integral of e to a function just equal e to the antiderivative of that function?
  3. Worked exampleFind the integral of x2 times ex^3 dx.
  4. Worked exampleFind the integral of the exponential function 5x dx.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4x2 ex^3, satisfying F(0) = 0.

Integrals That Produce a Logarithm

  1. How it worksHow does recognizing a numerator as the derivative of the denominator turn a fraction into a logarithm?
  2. Worked exampleFind the integral of (2x)/(x2 + 4) dx.
  3. Common mistakeWhy doesn't the power rule for integration ever apply to 1/x?
  4. Going furtherWhat happens to the antiderivative of 1/x once x is allowed to be negative?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4x/(x2 + 4), satisfying F(0) = 0.

Integrals Resulting in Inverse Trig Functions

  1. How it worksHow does matching an integrand's shape to 1/(1+x2) or 1/√(1-x2) point straight at an inverse trig antiderivative?
  2. Worked exampleFind the integral of 1/(9 + x2) dx.
  3. Worked exampleFind the integral of 1/√(4 - x2) dx.
  4. Common mistakeWhy isn't the integral of 1/(1 + x2) just the natural log of (1 + x2)?
  5. Going furtherWhat changes about the antiderivative when the denominator is a2 + x2 instead of 1 + x2?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 1/(x2 + 7), satisfying F(0) = 0.