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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Choose factors that simplify after differentiation.
- Justify the conclusion "The value is e−(e−1)=1" using the stated assumptions.
Before you start
Product rule and definite integrals.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Evaluate ∫₀¹x exp(x) dx.
Why this math matters
Choose factors that simplify after differentiation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Both factors are continuously differentiable on the interval.
- The definite integral uses bounds zero and one.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Reverse the product rule to integrate a product
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Evaluate ∫₀¹x exp(x) dx.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Choose u=x and dv=exp(x)dx, giving du=dx and v=exp(x)
Differentiating the polynomial lowers its degree.
Work through the mathematics
∫₀¹x exp(x)dx=[x exp(x)]₀¹−∫₀¹exp(x)dx
Integration by parts reverses the derivative of a product.
Check the conclusion
The value is e−(e−1)=1
Evaluate both the boundary term and the remaining integral.
The result
The value is e−(e−1)=1
Evaluate both the boundary term and the remaining integral.
Common mistakes to catch
- Keep the minus sign in the by-parts formula.
- The boundary term is evaluated at both endpoints.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find an antiderivative of x exp(x).
Show a hint
Keep the by-parts expression indefinite.
Reveal answer and explanation
(x−1)exp(x)+C
Differentiating gives exp(x)+(x−1)exp(x)=x exp(x).
Practice 2
Why not integrate the two factors separately and multiply?
Show a hint
Integration does not preserve products that way.
Reveal answer and explanation
It would violate the product rule
The extra subtraction term accounts for one factor's derivative.
Take the idea with you
Use integration by parts when differentiation simplifies one factor and the other is easy to integrate.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Turn an integral into an average height
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