← Calculus I

Unit 8: The Fundamental Theorem of Calculus and Techniques of Integration

The theorem tying every rectangle-counting limit back to antiderivatives, proved in full, plus the substitution toolkit for evaluating integrals fast.

9 topics · 35 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.

5.3 The Fundamental Theorem of Calculus, Part 1: The Accumulation Function

  1. How it worksWatch the accumulation function's own derivative trace out the exact height of the integrand underneath it.
  2. Common mistakeWhy isn't the accumulation function just a number, since it comes from an area under a curve?
  3. Worked exampleFind the derivative of A(x) = the integral of t2 from 0 to x.
  4. Worked exampleFind the derivative of A(x) = the integral of (3t + 1) from 1 to x.
  5. Ties togetherHow does the accumulation function turn every antiderivative search into a question about slopes instead of areas?
Your turn1 of 2
Let A(x) = the integral of (4t2 + 6) from 5 to x. Find A'(x) using the Fundamental Theorem, Part 1.

5.3 Proving the Fundamental Theorem of Calculus, Part 1

  1. How it worksProve that the derivative of the accumulation function A(x) equals f(x) using the Mean Value Theorem for Integrals.
  2. Worked exampleFind the derivative of A(x) = the integral of square root of (t2 + 1) from 2 to x using the Fundamental Theorem, Part 1.
  3. Worked exampleFind the derivative of A(x) = the integral of cos(t) from 0 to x2 using the Fundamental Theorem, Part 1, and the chain rule.
  4. Common mistakeWhy doesn't the Fundamental Theorem's first part require you to actually evaluate the integral to find the derivative?
Your turn1 of 2
Let A(x) = the integral of cos(t) from 0 to 4x2. Find A'(x) using the Fundamental Theorem, Part 1, and the chain rule.

5.3 The Fundamental Theorem of Calculus, Part 2: Evaluating Definite Integrals

  1. How it worksHow does plugging the two endpoints into any antiderivative and subtracting collapse an infinite sum into one calculation?
  2. Worked exampleFind the integral of sin(x) from 0 to π using the Fundamental Theorem of Calculus, Part 2.
  3. Worked exampleFind the integral of x2 from 1 to 3 using the Fundamental Theorem of Calculus, Part 2.
  4. Common mistakeWhy doesn't it matter which antiderivative you pick when evaluating a definite integral with the Fundamental Theorem?
  5. Worked exampleFind the integral of (3x2 - 2x + 1) from -1 to 2 using the Fundamental Theorem of Calculus.
Your turn1 of 2
Find the integral of (4x2 - 4x + 6) from -2 to 1.

5.4 The Net Change Theorem

  1. How it worksHow does integrating a rate function from a starting time to an ending time recover the total accumulated change?
  2. Worked exampleFind the total change in position of an object with velocity v(t) = 3t2 - 2t from t = 0 to t = 4.
  3. Common mistakeWhy isn't net change the same as total distance traveled when a rate switches sign in the middle of the interval?
Your turn1 of 2
Find the total change in position of an object with velocity v(t) = 4t2 - 6t from t = 0 to t = 6.

5.5 Integration by U-Substitution

  1. How it worksHow does choosing an inner function u and swapping dx for du in terms of u undo the chain rule backward?
  2. Worked exampleFind the integral of x·square root of (x2 + 9) dx using u-substitution.
  3. Common mistakeWhy doesn't every integral that looks like a product secretly hide a valid u-substitution?
  4. Worked exampleFind the integral of 2x·(x2 + 1)5 dx using u-substitution.
  5. Worked exampleFind the integral of cos(x)·sin4(x) dx using u-substitution.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4x·square root of (x2 + 6), satisfying F(0) = 0.

5.5 Substitution in a Definite Integral

  1. How it worksHow do you convert the original x-bounds into u-bounds so you never have to switch back to x at the end?
  2. Common mistakeWhy isn't it valid to substitute u back in for x at the end while still using the original x-bounds?
  3. Worked exampleFind the integral of x·(x2 + 1)3 dx from 0 to 2 using u-substitution with new bounds.
Your turn1 of 2
Find the integral of 4x·(x2 + 4)5 dx from 0 to 2 using u-substitution with new bounds.

5.6 Integrals Involving Exponential and Logarithmic Functions

  1. How it worksHow does substitution turn an integral like 1/x into the natural logarithm, with an absolute value tucked inside?
  2. Worked exampleFind the integral of (ln x)/x dx using u-substitution.
  3. Worked exampleFind the integral of e3x - 2/x dx.
  4. Common mistakeWhy isn't the integral of ln(x) itself simply 1/x, its own derivative in reverse?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 6·ln(x)/x, satisfying F(1) = 0.

5.7 Integrals Resulting in Inverse Trigonometric Functions

  1. How it worksHow do you match an integral's form to the derivative patterns of arcsin, arctan, and arcsec to reverse-engineer the antiderivative?
  2. Worked exampleFind the integral of 1/(1 + 9x2) dx.
  3. Worked exampleFind the integral of 1/(square root of (4 - x2)) dx.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4/(1 + (4x)2), satisfying F(0) = 0.

6.9 Integrals of the Hyperbolic Functions

  1. How it worksHow does reversing each hyperbolic derivative rule build a matching table of hyperbolic antiderivatives?
  2. Worked exampleFind the integral of sinh(2x) dx.
  3. Common mistakeWhy isn't the integral of cosh(x) equal to negative sinh(x)?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4sinh(4x), satisfying F(0) = 0.