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Teaching video
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Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Use a substitution consistently in a definite integral.
- Justify the conclusion "The integral is ∫₀¹exp(u)du=e−1" using the stated assumptions.
Before you start
Chain rule and elementary antiderivatives.
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 ∫₀¹2x exp(x²) dx.
Why this math matters
Use a substitution consistently in a definite integral. 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:
- The substitution is smooth and monotone on the intervals used.
- exp denotes the natural exponential.
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.
Change integral limits along with the variable
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Evaluate ∫₀¹2x exp(x²) 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
Set u=x², so du=2x dx
The derivative of the inner expression matches the remaining factor.
Work through the mathematics
x=0 gives u=0 and x=1 gives u=1
The bounds are translated into the new variable.
Check the conclusion
The integral is ∫₀¹exp(u)du=e−1
The antiderivative is evaluated entirely in the substituted variable.
The result
The integral is ∫₀¹exp(u)du=e−1
The antiderivative is evaluated entirely in the substituted variable.
Common mistakes to catch
- Do not mix original bounds with the substituted variable.
- Match the inner derivative before choosing a substitution.
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
Evaluate ∫₁²2x exp(x²) dx.
Show a hint
Convert the bounds to one and four.
Reveal answer and explanation
e⁴−e
The new upper bound is four, not two.
Practice 2
Why is replacing x² by u alone insufficient?
Show a hint
The differential changes too.
Reveal answer and explanation
The factor 2x dx must become du
A valid change of variables transforms both the integrand and the integration measure.
Take the idea with you
Reverse a chain-rule pattern to simplify an accumulated quantity.
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: Reverse the product rule to integrate a product
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