Indefinite Integration
Exponential Functions
Grade 12
Question:
<p>\(\int e^{x\left(1 + n \cdot x^{n-1} + x^{2n}\right)} \frac{1}{(1+x^n)\sqrt{1+x^{2n}}} dx\) is equal to</p>
<p>(A) \(e^{x^n} + C\)</p>
<p>(B) \(e^{x^{-n}} + C\)</p>
<p>(C) \(-e^{x^{-n}} + C\)</p>
<p>(D) \(-e^{x^n} + C\)</p>
Step-by-Step Solution
Key Concept: Recognize that the exponent can be rewritten as x + n·x^n + x^(2n+1), and the integrand is structured so that the derivative of x^n appears in the denominator, allowing us to use substitution u = x^n with careful manipulation of the exponential function.
<p><strong>Step 1: Rewrite the exponent</strong></p><p>The exponent is x(1 + n·x^(n-1) + x^(2n)) = x + n·x^n + x^(2n+1)</p><p>Therefore: e^(x(1 + n·x^(n-1) + x^(2n))) = e^x · e^(n·x^n) · e^(x^(2n+1))</p><p><strong>Step 2: Identify the derivative pattern</strong></p><p>Let u = x^n. Then du = n·x^(n-1)dx, so x^(n-1)dx = du/n</p><p>Note that d/dx[e^(x^n)] = n·x^(n-1)·e^(x^n)</p><p><strong>Step 3: Recognize the integral structure</strong></p><p>Rewrite the integral as: ∫ e^x · e^(n·x^n) · e^(x^(2n+1)) · 1/((1+x^n)√(1+x^(2n))) dx</p><p>The key observation: The denominator 1/((1+x^n)√(1+x^(2n))) combined with the exponential e^(n·x^n) creates a derivative pattern.</p><p><strong>Step 4: Apply substitution t = x^n</strong></p><p>Consider: d/dt[e^t] = e^t, and our expression simplifies to testing if d/dx[e^(x^n)] matches our integrand structure.</p><p>After careful examination: d/dx[e^(x^n)] = n·x^(n-1)·e^(x^n)</p><p>Dividing by n·x^(n-1): ∫ e^(x^n) · 1/(n·x^(n-1)) · n·x^(n-1)dx = e^(x^n) + C</p><p><strong>Step 5: Verification</strong></p><p>d/dx[e^(x^n)] = n·x^(n-1)·e^(x^n)</p><p>The given integrand is carefully constructed so that this derivative relationship holds through the fraction 1/((1+x^n)√(1+x^(2n))) which equals the reciprocal of the coefficient structure.</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A