Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>If \(\int \dfrac{(\sqrt{x})^5}{(\sqrt{x})^7 + x^6}\, dx = \lambda \ln\left(\dfrac{x^a}{x^a+1}\right) + c\), then \(a + \lambda\) is</p>
<p>\(= 2\)</p>
<p>\(> 2\)</p>
<p>\(< 2\)</p>
<p>\(= 1\)</p>
Step-by-Step Solution
Key Concept: Simplify the integrand by expressing everything in terms of a single power of x, then use substitution u = x^a to reduce it to a standard logarithmic form.
<p><strong>Step 1:</strong> Rewrite the integrand using fractional exponents.</p><p>∫(x^(5/2))/(x^(7/2) + x^6) dx</p><p><strong>Step 2:</strong> Factor the denominator: x^(7/2) + x^6 = x^(7/2)(1 + x^(5/2))</p><p>∫(x^(5/2))/(x^(7/2)(1 + x^(5/2))) dx = ∫(1)/(x^1(1 + x^(5/2))) dx = ∫(1)/(x(1 + x^(5/2))) dx</p><p><strong>Step 3:</strong> Use substitution u = x^(5/2), so du = (5/2)x^(3/2) dx. Note that x^(5/2) = u means x^(1) = u^(2/5).</p><p>Alternatively, substitute t = x^(5/2): dt = (5/2)x^(3/2) dx, which gives dx = (2/5)x^(-3/2) dt.</p><p>This leads to: (2/5)∫(1)/(t(1 + t)) dt</p><p><strong>Step 4:</strong> Use partial fractions: (1)/(t(1 + t)) = (1/t) - (1/(1 + t))</p><p>(2/5)[∫(1/t) dt - ∫(1/(1 + t)) dt] = (2/5)[ln|t| - ln|1 + t|] = (2/5)ln(t/(1 + t))</p><p><strong>Step 5:</strong> Substitute back t = x^(5/2):</p><p>= (2/5)ln(x^(5/2)/(x^(5/2) + 1)) + c</p><p><strong>Step 6:</strong> Compare with λ ln(x^a/(x^a + 1)) + c:</p><p>λ = 2/5 and a = 5/2</p><p>∴ a + λ = 5/2 + 2/5 = 25/10 + 4/10 = <strong>29/10</strong></p>
Correct Answer: A