Steply

The Fundamental Theorem, part 1

Calculus 1 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. The rule
  2. Put x in place of t
  3. Check

The steps for Find d/dx ∫ from 1 to x of (t^2 + 1) dt. Every solution names its steps, so you always know what comes next.

Worked examples

Find d/dx ∫ from 1 to x of (t^2 + 1) dt Easy

The question

Question: find the derivative of ∫ from 1 to x of (t² + 1) dt.

ddx∫ from 1 to x of (t2 + 1) dt =

1 The rule

The rule: the derivative of the area from a to x is the function at x.

Just put x in place of t.

ddx∫ from a to x of f(t) dt = f(x)

2 Put x in place of t

Put x in place of t.

So the derivative is x² + 1.

ddx∫ from 1 to x of (t2 + 1) dt = x2 + 1

Careful!

There is nothing to take away.

The derivative of the area is just f at x.

The bottom number 1 does not matter.

Careful: not t2 + 1 − f(1) ✗

3 Check

Let's check at x = 2.

Our answer gives 5.

answer at x = 2: 5

Now the slope of the area function F between x = 2 and 2.001, with a calculator: about 5.002.

The same!

Right!

F(2.001) − F(2)0.001 ≈ 5.002

Let's look back.

First: the rule.

Then: put x in place of t.

Last: check.

Answer
Careful: not t2 + 1 − f(1) ✗

▶ Watch it solved

Find d/dx ∫ from 0 to x of 1/(t^2 + 1) dt Medium

The question

Question: find the derivative of ∫ from 0 to x of 1/(t² + 1) dt.

ddx∫ from 0 to x of 1t2 + 1 dt =

1 The rule

The rule: the derivative of the area from a to x is the function at x.

Just put x in place of t.

ddx∫ from a to x of f(t) dt = f(x)

2 Put x in place of t

Put x in place of t.

So the derivative is 1/(x² + 1).

ddx∫ from 0 to x of 1t2 + 1 dt = 1x2 + 1

Careful!

There is nothing to take away.

The derivative of the area is just f at x.

The bottom number 0 does not matter.

Careful: not 1t2 + 1 − f(0) ✗

3 Check

Let's check at x = 1.

Our answer gives 0.5.

answer at x = 1: 0.5

Now the slope of the area function F between x = 1 and 1.001, with a calculator: about 0.5.

The same!

Right!

F(1.001) − F(1)0.001 ≈ 0.5

Let's look back.

First: the rule.

Then: put x in place of t.

Last: check.

Answer
Careful: not 1t2 + 1 − f(0) ✗

▶ Watch it solved

Find d/dx ∫ from 2 to x^2 of 1/t dt Harder

The question

Question: find the derivative of ∫ from 2 to x² of 1/t dt.

ddx∫ from 2 to x² of 1t dt =

1 The rule

The rule: the derivative of the area from a to x is the function at x.

Just put x in place of t.

ddx∫ from a to x of f(t) dt = f(x)

2 Put the top in place of t

The top is not just x: it is x².

Call it u.

Its slope is u′.

top: u = x2, u′ = 2x

The chain rule: put u in place of t, then multiply by u′.

ddx∫ from 2 to x² of 1t dt = f(u) × u′

Put x² in place of t, times 2x.

= 1x2 × 2x

Careful!

Do not stop at f of u.

The top is x², so we must multiply by its slope, 2x.

Careful: not 1x2 ✗

Multiply it out: 2 over x.

= 2x

3 Check

Let's check at x = 3.

Our answer gives 0.6667.

answer at x = 3: 0.6667

Now the slope of the area function F between x = 3 and 3.001, with a calculator: about 0.667.

The same!

Right!

F(3.001) − F(3)0.001 ≈ 0.667

Let's look back.

First: the rule.

Then: put the top in place of t.

Last: check.

Answer
= 2x

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. d/dx ∫ from 1 to x of (t^2 + 1) dt
    Answer
    Careful: not t2 + 1 − f(1) ✗
  2. d/dx ∫ from 0 to x of t^3 dt
    Answer
    Careful: not t3 − f(0) ✗
  3. d/dx ∫ from 2 to x of (3t^2 - t) dt
    Answer
    Careful: not 3t2 − x − f(2) ✗
  4. d/dx ∫ from 0 to x of cos(t) dt
    Answer
    Careful: not cos x − f(0) ✗
  5. d/dx ∫ from 1 to x of e^t dt
    Answer
    Careful: not et − f(1) ✗
  6. d/dx ∫ from 1 to x of 1/t dt
    Answer
    Careful: not 1t − f(1) ✗
  7. d/dx ∫ from 0 to x of √(1 + t^3) dt
    Answer
    Careful: not √1 + t3 − f(0) ✗
  8. find F′(x) if F(x) = ∫ from 0 to x of √(1 + t^3) dt
    Answer
    Careful: not √1 + t3 − f(0) ✗

Medium

  1. d/dx ∫ from 0 to x of 1/(t^2 + 1) dt
    Answer
    Careful: not 1t2 + 1 − f(0) ✗
  2. d/dx ∫ from 1 to x of 1/(t^3 + 1) dt
    Answer
    Careful: not 1t3 + 1 − f(1) ✗
  3. d/dx ∫ from 0 to x of √(t^2 + 4) dt
    Answer
    Careful: not √t2 + 4 − f(0) ✗
  4. d/dx ∫ from x to 5 of cos(t) dt
    Answer
    = −cos x
  5. d/dx ∫ from x to 3 of (t^2 + 2) dt
    Answer
    = −(x2 + 2)
  6. d/dx ∫ from x to 1 of √(t^4 + 1) dt
    Answer
    = −√x4 + 1

Harder

  1. d/dx ∫ from 2 to x^2 of 1/t dt
    Answer
    = 2x
  2. d/dx ∫ from 0 to x^3 of sin(t) dt
    Answer
    = 3x² sin(x3)
  3. d/dx ∫ from 0 to x^2 of (t + 1) dt
    Answer
    = 2x3 + 2x
  4. d/dx ∫ from 1 to x^2 of t^2 dt
    Answer
    = 2x5
  5. d/dx ∫ from 0 to 3x of cos(t) dt
    Answer
    = 3 cos(3x)
  6. d/dx ∫ from 0 to x^2 of √(1 + t^2) dt
    Answer
    = 2x√1 + x4
  7. d/dx ∫ from 0 to 2x of e^t dt
    Answer
    = 2 e2x
  8. d/dx ∫ from 1 to x^3 of 1/(t^2 + 1) dt
    Answer
    = 3x²x6 + 1