The fundamental theorem and substitution

Calculus I. Also known as: math 151, math 2413, math 1550, math 1910, math 140, calc 1, calc i, calculus 1, calculus i, differential calculus, calculus for science and engineering, math 121

Link derivatives and integrals with the fundamental theorem, then use substitution to integrate composite functions.

The fundamental theorem, part 1

Define A(x) as the accumulated area under f from a fixed starting point a up to x. The first part of the fundamental theorem says that the derivative of A is the original function: d/dx of the integral of f(t) dt from a to x equals f(x), when f is continuous. Differentiation undoes accumulation. If the upper limit is a function g(x), the chain rule adds a factor: the derivative is f(g(x)) * g'(x).

Example: d/dx of the integral of cos(t) dt from 1 to x^3 is cos(x^3) * 3x^2.

The fundamental theorem, part 2

If F is any antiderivative of f, then the integral of f from a to b equals F(b) - F(a). This turns area problems into algebra: find an antiderivative, evaluate at the top limit, subtract the value at the bottom limit. The constant C cancels, so you can leave it out.

Substitution

Substitution reverses the chain rule. When you see a function and its derivative (up to a constant) in the integrand, let u be the inside function.

  1. Choose u, for example the inside of a power or an exponent.
  2. Compute du = u' dx and rewrite everything in terms of u.
  3. Integrate in u.
  4. Substitute back, or for a definite integral convert the limits to u-values and do not substitute back.

Worked example

Evaluate the integral of 6x(x^2 + 2)^2 dx from x = 0 to x = 1.

Let u = x^2 + 2, so du = 2x dx and 6x dx = 3 du. When x = 0, u = 2, and when x = 1, u = 3. The integral becomes 3 times the integral of u^2 du from 2 to 3, which is u^3 evaluated from 2 to 3: 27 - 8 = 19.

Another example: the integral of 2x e^(x^2) dx. With u = x^2, du = 2x dx, so the integral is e^u + C = e^(x^2) + C.

Average value

The average value of f on [a, b] is (1/(b - a)) times the integral of f from a to b. For f(x) = x on [0, 4] the average is (1/4)(8) = 2.

Common mistakes

  • Forgetting the g'(x) factor when the upper limit is a function of x.
  • Forgetting to convert the limits when substituting in a definite integral.
  • Leaving leftover x terms in the integrand after substitution; if x remains, the choice of u may be wrong.
  • Reversing F(b) - F(a).

Practice questions

  1. What is the derivative with respect to x of the integral of sin(t^2) dt from t = 0 to t = x?

  2. What is the derivative with respect to x of the integral of sqrt(1 + t) dt from t = 1 to t = x^2?

  3. Using the fundamental theorem, what is the integral of 1/x from x = 1 to x = e?

  4. What is the integral of 2x(x^2 + 5)^3 with respect to x?

  5. What is the integral of x*e^(x^2) with respect to x?

  6. What is the integral of x/(x^2 + 1) from x = 0 to x = 1?