Calculus Solution Chapter 10.github.com Ctzhou86 -

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• Copy the solutions directly into your homework without trying the problem first.
• Use the solutions as a substitute for reading your textbook or attending lectures.

• Integration by parts: choose u to simplify derivative and dv to integrate cleanly. For repeated parts, use tabular integration.
• Trigonometric integrals: for sin^m x cos^n x, use substitution when one power is odd (save one factor and use u = sin x or cos x); use identities when both even (power‑reducing).
• Trig substitution: for sqrt(a^2 − x^2) use x = a sin θ; for sqrt(a^2 + x^2) use x = a tan θ; for sqrt(x^2 − a^2) use x = a sec θ. Convert integrand and dx, simplify, evaluate, revert to x.
• Partial fractions: factor denominator over reals (linear/quadratic). Decompose into sum of simpler rational functions, integrate termwise (linear -> ln, irreducible quadratic -> arctan).

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