Definite Integration
Grade None
Question:
<p><span class="math-tex">\(\int_{0}^{\pi / 4} \frac{\cos ^{2} x \sin ^{2} x}{\left(\cos ^{3} x+\sin ^{3} x\right)^{2}} d x\)</span> is equal to</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{9}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{12}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{6}\)</span></p>
Step-by-Step Solution
Key Concept: Divide the numerator and denominator by cos^6 x to transform the integrand into a function of tan x and its derivative sec^2 x, enabling a simple substitution.
<p>Dividing numerator and denominator by <span class="math-tex">\(\cos x\)</span><br />
<span class="math-tex">\(\int_{0}^{\pi / 4} \frac{\tan ^{2} x \sec ^{2} x {dx}}{\left(1+\tan ^{3} x\right)^{2}} {dx}\)</span><br />
Let <span class="math-tex">\(1+\tan ^{3} x={t}\)</span><br />
<span class="math-tex">\(\Rightarrow \tan ^{2} x \sec ^{2} x d x=\frac{d t}{3}\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{1}{3} \int_{1}^{2} \frac{d t}{t^{2}}=\frac{1}{6}\)</span></p>
Correct Answer: D