What is the derivative with respect to x of ∫ from 1 to x of (t^3 +… Step by Step

Step by step

The question

Question: find the derivative of ∫ from 1 to x of (t³ + 1) dt.

ddx∫ from 1 to x of (t³ + 1) dt

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The rule

The rule: the derivative of the area from a to x is the function inside the integral, 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 x³ + 1.

ddx∫ from 1 to x of (t³ + 1) dt = x³ + 1

Careful!

The derivative of the area is just f at x.

The bottom number 1 does not matter.

Careful: the bottom number 1 does not matter

3 Check

Let's check at x = 2.

Our answer gives 9.

answer at x = 2: 9

Now the slope of the area function F, the integral itself, between x = 2 and 2.001, with a calculator: about 9.006.

The same!

Right!

F(2.001) − F(2)0.001 ≈ 9.006

4 Pick the choice

Our answer is x³ + 1.

That is choice A.

A) x³ + 1

5 Why not the others

B is x⁴ over 4 + x.

Wrong: integrated only: the derivative undoes it.

B) x⁴/4 + x ✗ integrated only: the derivative undoes it

C is 3x².

Wrong: differentiated t³ + 1.

C) 3x² ✗ differentiated t^3 + 1

D is x³.

Wrong: lost the + 1.

D) x³ ✗ lost the + 1

Let's look back.

First: the rule.

Then: put x in place of t.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer