A-level · Beta

Substitution and integration by parts

Link a change of variable or a product-rule reversal to a definite integral.

Choose a method, change the parameters, then reveal each step and check the shaded signed area.

∫01(2x+(1))2 dx\int_{0}^{1}(2x+(1))^{2}\,dx
Choose a method, change the parameters, then reveal each step and check the shaded signed area.-1-0.9-0.251.80.54.51.257.229.9xy

Shading is signed area: parts below the axis contribute negatively.

Signed integral (approx.)
4.3333
Method
Reverse the chain rule
Steps revealed
0 of 3

    Bounds are ordered from lower to upper for this model. The method changes the algebra; it does not change the integral. Displayed numerical values are rounded.

    Predict, test and explain

    Step 1 of 3

    Predict

    If u = 2x + 1, what replaces dx?

    InvestigateWhen u = ax + b, what replaces dx?

    Explore and compare

    Choose the investigation above. Predict what will change before you move a control.

    Try it in the simulation. Change one thing at a time.

    Notes and saved values stay in this tab. They disappear when you reload.

    Hints, self-check and connections
    Try a self-check

    Fixed values, separate from the diagram controls.

    If u = 2x + 1, what replaces dx?

    ∫01(2x+1)2 dx\int_0^1(2x+1)^2\,dx

    Hint

    Check an antiderivative by differentiating it.

    Reveal answer

    Since du = 2dx, dx = du/2. The bounds must also be expressed in u.

    133\frac{13}{3}

    Build an explanation

    Hint 1

    Check an antiderivative by differentiating it.

    Hint 2

    Since du = 2dx, dx = du/2. The bounds must also be expressed in u.

    Hint 3

    Use the model to check a prediction, then explain the result.

    Worked example

    This example uses fixed values, separate from the diagram controls.

    Evaluate ∫₀¹(2x + 1)² dx.

    1. Set u = 2x + 1, so dx = du/2.
    2. The new bounds are 1 and 3.
    3. Evaluate ½[u³/3] from 1 to 3.

    Answer: 13/3.

    Connect this idea

    Watch out: Since du = 2dx, dx = du/2. The bounds must also be expressed in u.

    Specification and learning route

    AQA H5 · Edexcel Pure 8.5

    A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

    Useful starting knowledge: Standard integrals; Chain and product rules; Substitution in algebra; Partial fractions when used.

    AQA specification · Edexcel specification

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