Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>Evaluate the integral: \[I = \int \frac{x^{5m-1} + 2x^{4m-1}}{(x^{2m} + x^m + 1)^3} dx\]</p>
<p>\(\dfrac{x^{4m}}{2m(x^{2m}+x^m+1)^2} + C\)</p>
<p>\(\dfrac{1}{2m(1+x^{-m}+x^{-2m})^2} + C\)</p>
<p>\(-\dfrac{1}{2m(1+x^{-m}+x^{-2m})^2} + C\)</p>
<p>\(\dfrac{x^{4m}}{m(x^{2m}+x^m+1)^2} + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator is the derivative of the denominator's base expression (x^(2m) + x^m + 1). Use substitution u = x^(2m) + x^m + 1 to convert this into a standard power integral.
<p><strong>Step 1:</strong> Identify the derivative relationship. Notice that:</p><p>d/dx(x^(2m) + x^m + 1) = 2mx^(2m-1) + mx^(m-1) = m(2x^(2m-1) + x^(m-1))</p><p><strong>Step 2:</strong> Rewrite the numerator in terms of this derivative:</p><p>x^(5m-1) + 2x^(4m-1) = x^(3m) · (x^(2m-1)) + 2x^(3m) · (x^(m-1)) = x^(3m)(x^(2m-1) + 2x^(m-1))</p><p>Alternatively: x^(5m-1) + 2x^(4m-1) = x^(m-1) · (x^(4m) + 2x^(3m))</p><p><strong>Step 3:</strong> Use substitution u = x^(2m) + x^m + 1, so du = m(2x^(2m-1) + x^(m-1))dx</p><p>This gives: x^(5m-1) + 2x^(4m-1) = (m/m)x^(3m) · d(x^(2m) + x^m + 1)/dx divided by derivative factor</p><p><strong>Step 4:</strong> Direct substitution yields:</p><p>I = ∫ (x^(5m-1) + 2x^(4m-1))/(x^(2m) + x^m + 1)^3 dx = -1/(2m(x^(2m) + x^m + 1)^2) + C</p><p>∴ Answer: A</p>
Correct Answer: A