Question

\[ \int \frac{(2x^{12} + 5x^9)}{(1 + x^3 + x^5)^3} \, dx \] equals:

• A. \(\frac{x^{10}}{2(x^5 + x^3 + 1)^2} + C\)
• B. \(\frac{x^2 + 2x}{(x^5 + x^3 + 1)} + C\)
• C. \(\log(x^5 + x^3 + 1 + \sqrt{2x^{12}+5x^9}) + C\)
• D. None of the above

Hint:

When the integrand resembles the derivative of a rational function, try differentiating the options.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Recognize the derivative of the denominator.

Step 2: Detailed Explanation:
Let \(t = x^5 + x^3 + 1\). Then \(dt = (5x^4 + 3x^2)dx\). The numerator \(2x^{12} + 5x^9 = x^9(2x^3 + 5)\). Try to express in terms of \(t\) derivative. Multiply numerator and denominator: Consider \(d(1/t^2) = -2/t^3 dt\). So \(\int \frac{2x^{12}+5x^9}{(1+x^3+x^5)^3} dx = \int \frac{2x^{12}+5x^9}{t^3} dx\). Also, \(x^{10} dx\) appears in derivative of \(x^{11}\)? Better approach: Let \(u = x^5\). Then \(du = 5x^4 dx\). Rewrite integral. Alternatively, note that derivative of \(\frac{x^{10}}{(x^5+x^3+1)^2}\) gives the integrand. Checking: Differentiate option (A): \(d/dx \left[\frac{x^{10}}{2(x^5+x^3+1)^2}\right]\). Using quotient rule, we get \(\frac{10x^9(x^5+x^3+1)^2 - x^{10} \cdot 2(x^5+x^3+1)(5x^4+3x^2)}{2(x^5+x^3+1)^4} = \frac{10x^9(x^5+x^3+1) - 2x^{10}(5x^4+3x^2)}{2(x^5+x^3+1)^3} = \frac{10x^9 x^5 + 10x^9 x^3 + 10x^9 - 10x^{14} - 6x^{12}}{2(x^5+x^3+1)^3} = \frac{10x^{14}+10x^{12}+10x^9 - 10x^{14} - 6x^{12}}{2(x^5+x^3+1)^3} = \frac{4x^{12}+10x^9}{2(x^5+x^3+1)^3} = \frac{2x^{12}+5x^9}{(x^5+x^3+1)^3}\). Yes, matches.

Step 3: Final Answer:
Option (A) is correct.