Substitution

Learn substitution with a nonlinear input, substitution with a linear input.

Lesson 2 of 32 parts Practice Flashcards

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Substitution with a Nonlinear Input

Objective: Explain substitution with a nonlinear input, apply the stated method, and check its conditions.

Substitution reverses the chain rule by setting u=g(x) and replacing g′(x)dx with du. In a definite integral, transform the bounds into u values. Every remaining expression must use the new variable until the calculation is complete. Do not mix old x bounds with an integrand written in u.

Formulas

$u=g(x), du=g′(x)dx$

Worked examples

Try itEvaluate ∫₀^7 2x(x²+7) dx.Show answer

Step 1: Let u=x²+b so du=2x dx. Step 2: New bounds are 7 and 56; integrate u du. Step 3: The value is [(u²)/2] from 7 to 56, or 3087/2.

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Substitution with a Nonlinear Input

Objective: Explain substitution with a nonlinear input, apply the stated method, and check its conditions.

Formulas

$u=g(x), du=g′(x)dx$

Worked examples

Try itEvaluate ∫₀^7 2x(x²+7) dx.Show answer

Step 1: Let u=x²+b so du=2x dx. Step 2: New bounds are 7 and 56; integrate u du. Step 3: The value is [(u²)/2] from 7 to 56, or 3087/2.

Substitution with a Linear Input

Objective: Explain substitution with a linear input, apply the stated method, and check its conditions.

For u=ax+b with a≠0, du=a dx. A matching factor a in the integrand can be absorbed into du; otherwise dx=du/a contributes a scale factor. Map both original endpoints through u=ax+b when evaluating a definite integral. Account for the derivative of the substituted expression.

Formulas

$u=ax+b; dx=du/a$

Worked examples

Try itEvaluate ∫₀^7 6(6x+7)² dx.Show answer

Step 1: Let u=ax+b so du=a dx. Step 2: New bounds are 7 and 49. Step 3: Integrate u² du to get 39102.

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