Indefinite Integration
Basic Integration Rules
Grade 12
Question:
<p>Evaluate <span class="math">\(\int 2^x (2012)^{\sin^{-1}(2012)} \sin x\, dx\)</span></p>
<p>(a) <span class="math">\((\log_{2012} e)(2012)^{\sin^{-1}(2012)} + C\)</span></p>
<p>(b) <span class="math">\((\log_{2012} e)(2012)^{\sin^{-1}(2012)x} + C\)</span></p>
<p>(c) <span class="math">\((\log_{2012} e)(2012)^{\sin^{-1}(2012)x} + C\)</span></p>
<p>(d) <span class="math">\(\frac{(2012)^{\sin^{-1}(2012)x}}{2(\log_{2012} e)} + C\)</span></p>
Step-by-Step Solution
Key Concept: Identify constants versus variables in the integrand; (2012)^{sin^{-1}(2012)} is constant while 2^x is the variable part.
<p>Recognize that <span class="math">$(2012)^{\sin^{-1}(2012)}$</span> is a constant. The integral becomes <span class="math">$(2012)^{\sin^{-1}(2012)} \int 2^x \sin x\, dx$</span>.</p>
Correct Answer: A