Integrating Composite Functions (Leaving Cert Applied Maths): Quizzes

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Integrating Composite Functions
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Why is substitution needed for composite functions in integration?

No reverse chain rule exists

What is the main aim of the substitution method?

Transform to simpler function in tables

What is the most critical step in the substitution process?

Selecting right substitution at start

What makes a good substitution choice?

Derivative appears naturally in integrand

For ∫013x2x3+2dx\int_0^1 \frac{3x^2}{x^3+2} dx, which substitution is best?

u=x3+2u = x^3 + 2

Why is u=x3+2u = x^3 + 2 the best choice for ∫013x2x3+2dx\int_0^1 \frac{3x^2}{x^3+2} dx?

3x2dx3x^2 dx appears in the integral

What does ∫1udu\int \frac{1}{u} du equal?

ln⁡u\ln u

What are the two valid approaches for definite integrals?

Change limits OR substitute back

Do both methods for definite integrals give the same result?

Yes, both give identical results

Where is the inner function usually found?

Inside brackets, roots, or denominator

For ∫12xx+1dx\int_1^2 \frac{x}{x+1} dx with u=x+1u = x+1, what is xx in terms of uu?

x=u−1x = u - 1

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