Indefinite Integration
Integration of Exponential Functions
Grade 12
Question:
<p>\(\int \frac{2^x + 3^x}{5^x}\) dx is equal to</p>
<p>(a) \(\frac{2^x}{\log 2} + \frac{3^x}{\log 3} - \frac{5^x}{\log 5} + C\)</p>
<p>(b) \(\frac{(25/5)^x}{x\log_e(25/5)} + \frac{(35/5)^x}{x\log_e(35/5)} + C\)</p>
<p>(c) \(\frac{(2/5)^x}{\log_e(2/5)} + \frac{(3/5)^x}{\log_e(3/5)} + C\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Rewrite the integrand as a sum of exponential functions with fractional bases, then apply the standard exponential integral formula.
<p><strong>Solution:</strong></p><p>$\int \frac{2^x + 3^x}{5^x}$ dx = $\int \left(\frac{2^x}{5^x} + \frac{3^x}{5^x}\right)$ dx</p><p>= $\int \left(\left(\frac{2}{5}\right)^x + \left(\frac{3}{5}\right)^x\right)$ dx</p><p>= $\frac{(2/5)^x}{\ln(2/5)} + \frac{(3/5)^x}{\ln(3/5)} + C$</p><p>∴ Answer is (c).</p>
Correct Answer: C