Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>If \(\int \dfrac{\cos^4 x\,dx}{(\sin^3 x)(\sin^5 x + \cos^5 x)^{3/5}}\), and the result involves constants \(A\) and \(B\) such that \(A = 5\) and \(B = \dfrac{2}{5}\), find \(AB\).</p>
Step-by-Step Solution
Key Concept: Divide numerator and denominator by sin⁸x to convert the integrand into a form involving (1 + cot⁵x), then use substitution u = cot x to reduce to a standard power form.
<p><strong>Step 1:</strong> Divide numerator and denominator by sin⁸x:</p><p>∫ cos⁴x/(sin³x(sin⁵x + cos⁵x)^(3/5)) dx = ∫ (cos⁴x/sin⁸x)/((sin⁵x + cos⁵x)^(3/5)/sin⁸x) dx</p><p>= ∫ cot⁴x/(1 + cot⁵x)^(3/5) · csc²x dx</p><p><strong>Step 2:</strong> Recognize that the denominator becomes (sin⁵x + cos⁵x)^(3/5)/sin⁸x = ((sin⁵x + cos⁵x)/sin⁵x)^(3/5) · sin^(-3/5)x · sin³x = (1 + cot⁵x)^(3/5) · csc²x in the measure.</p><p><strong>Step 3:</strong> Let u = 1 + cot⁵x, then du = -5cot⁴x·csc²x dx</p><p>So cot⁴x·csc²x dx = -du/5</p><p><strong>Step 4:</strong> The integral becomes:</p><p>∫ -1/(5u^(3/5)) du = -1/5 · u^(2/5)/(2/5) = -(1/5) · (5/2) · u^(2/5) = -1/2 · (1 + cot⁵x)^(2/5)</p><p><strong>Step 5:</strong> From the solution form: A = 5 (coefficient in denominator after du), B = 2/5 (exponent coefficient)</p><p>∴ AB = 5 × (2/5) = 2</p>
Correct Answer: 2