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Integration by Substitution

Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand.

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Student-friendly explanation

Substitution is useful when the integrand contains a composite expression and its derivative, up to a constant factor. After integrating in u, the answer must be written back in terms of x for an indefinite integral.

How to write this in exams

  1. 1

    Start with the exact idea

    Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand.

  2. 2

    Then show how to use it

    Choose the inner expression as u. Differentiate it to find du. Rewrite all x-dependent parts and dx in terms of u. Integrate using a standard formula. Convert back to x for indefinite integrals. For definite integrals, handle limits consistently.

  3. 3

    Add one concrete example

    For ∫2x cos(x^2) dx, take u=x^2, so du=2x dx. The integral becomes ∫cos u du=sin u+C=sin(x^2)+C.

  4. 4

    Avoid this incomplete answer

    Taking u=x^2+5 in ∫x√(x^2+5) dx but writing x dx=du instead of du/2, causing the answer to be twice the correct value.

Definition

Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand.

Example

For ∫2x cos(x^2) dx, take u=x^2, so du=2x dx. The integral becomes ∫cos u du=sin u+C=sin(x^2)+C.

Rule to remember

Method condition: if the integral has the form ∫f(g(x))g'(x) dx, put u=g(x), du=g'(x) dx, then integrate ∫f(u) du and substitute back. For definite integrals, either change limits to u-values or substitute back before applying original limits.

Memory hook

Substitution is the chain rule walked backward.

Examples and method

Worked example

Evaluate ∫x√(x^2+5) dx. Let u=x^2+5, so du=2x dx and x dx=du/2. Integral =(1/2)∫u^(1/2) du=(1/2)·(2/3)u^(3/2)+C=(1/3)(x^2+5)^(3/2)+C.

Method to apply

Choose the inner expression as u. Differentiate it to find du. Rewrite all x-dependent parts and dx in terms of u. Integrate using a standard formula. Convert back to x for indefinite integrals. For definite integrals, handle limits consistently.

Diagram support

No diagram is required. The key representation is the chain rule in reverse: derivative of the inside expression must be accounted for.

How CBSE asks it

Appears in short-answer and long-answer questions with algebraic, exponential, logarithmic, and trigonometric composite functions. Definite integral versions may test correct transformation of limits.

Avoid common mistakes

Common confusion

A common error is choosing u correctly but forgetting to replace dx and the remaining factor, leaving a mixture of u and x.

Common wrong answer

Taking u=x^2+5 in ∫x√(x^2+5) dx but writing x dx=du instead of du/2, causing the answer to be twice the correct value.

Exam tip

Look for an inside function whose derivative is also present. If the derivative is present with a constant multiplier, substitution is likely efficient.

Quick check

Evaluate ∫3(3x+1)^4 dx.

Let u=3x+1, du=3 dx. Integral =∫u^4 du=u^5/5+C=(3x+1)^5/5+C.

Answer writing and exam use

1-mark answer

Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand.

2-mark answer

Integration by substitution changes a difficult integral into a standard one by putting u=g(x), so that du=g'(x) dx matches part of the integrand. Method condition: if the integral has the form ∫f(g(x))g'(x) dx, put u=g(x), du=g'(x) dx, then integrate ∫f(u) du and substitute back. For definite integrals, either change limits to u-values or substitute back before applying original limits. For ∫2x cos(x^2) dx, take u=x^2, so du=2x dx. The integral becomes ∫cos u du=sin u+C=sin(x^2)+C.

3-mark answer

Substitution is useful when the integrand contains a composite expression and its derivative, up to a constant factor. After integrating in u, the answer must be written back in terms of x for an indefinite integral. Method condition: if the integral has the form ∫f(g(x))g'(x) dx, put u=g(x), du=g'(x) dx, then integrate ∫f(u) du and substitute back. For definite integrals, either change limits to u-values or substitute back before applying original limits. Evaluate ∫x√(x^2+5) dx. Let u=x^2+5, so du=2x dx and x dx=du/2. Integral =(1/2)∫u^(1/2) du=(1/2)·(2/3)u^(3/2)+C=(1/3)(x^2+5)^(3/2)+C. Appears in short-answer and long-answer questions with algebraic, exponential, logarithmic, and trigonometric composite functions. Definite integral versions may test correct transformation of limits. Taking u=x^2+5 in ∫x√(x^2+5) dx but writing x dx=du instead of du/2, causing the answer to be twice the correct value.
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