Unit 4 · Lesson 5 of 5

Reverse the chain rule with a new variable.

A composed rate law can become simpler when its inner quantity is treated as the integration variable.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its definite integral.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase upper endpoint b? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

Integrand

Upper endpoint b1

definite integral

u=x^2,\quad du=2x\,dx,\quad I=e^{1}-1=1.71828

Changing variables includes the differential and integration limits; do not mix u bounds with an x integrand.

Assumptions, units, and limits

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

Choose an inner function whose derivative appears as a factor. Replace both integrand and differential, transform the endpoint values for a definite integral, and integrate in one consistent variable.

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Keep the reasoning visible

u=g(x),\quad du=g'(x)\,dx

Changing variables includes the differential and integration limits; do not mix u bounds with an x integrand.

What makes this model useful?

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

A variable change without changing the differential or bounds mixes two coordinate descriptions.

A worked example

Calculate the starting scenario

Use upper endpoint b = 1 .
$u=x^2,\quad du=2x\,dx,\quad I=e^{1}-1=1.71828$
Changing variables includes the differential and integration limits; do not mix u bounds with an x integrand. The checked quantity is definite integral.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the definite integral using upper endpoint b 1 . State the meaning of its sign and units.

Checked quantity: definite integral · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

Integrand

Upper endpoint b1

definite integral

u=x^2,\quad du=2x\,dx,\quad I=e^{1}-1=1.71828

Changing variables includes the differential and integration limits; do not mix u bounds with an x integrand.

Assumptions, units, and limits

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Calculate and explain

Apply the idea to a new scenario

With $b=0.5$ find the integral of 2x exp(x²) from 0 to b.

Checked quantity: definite integral · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

Integrand

Upper endpoint b0.5

definite integral

u=x^2,\quad du=2x\,dx,\quad I=e^{0.25}-1=0.284025

Changing variables includes the differential and integration limits; do not mix u bounds with an x integrand.

Assumptions, units, and limits

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Read the model

\int_0^b2xe^{x^2}\,dx=\int_0^{b^2}e^u\,du=e^{b^2}-1

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

Integrand

Upper endpoint b1

definite integral

u=x^2,\quad du=2x\,dx,\quad I=e^{1}-1=1.71828

Changing variables includes the differential and integration limits; do not mix u bounds with an x integrand.

Assumptions, units, and limits

Let $u=x^2$ and transform both differential and bounds. This is the chain rule run backward.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Choose an inner function whose derivative appears as a factor. Replace both integrand and differential, transform the endpoint values for a definite integral, and integrate in one consistent variable.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.