NCERT Class 12 · Ch 7 Exercise 7.2 · Q38
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.2 Question 38
Evaluate \(\int \frac{10^{x} \log{\left(10 \right)} + 10 x^{9}}{10^{x} + x^{10}}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: D: \(\log{\left(10^{x} + x^{10} \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- The numerator is the derivative of the strictly positive denominator. Use ∫g′/g=log|g|.
- An antiderivative is F(x)=\(\log{\left(10^{x} + x^{10} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{10^{x} \log{\left(10 \right)} + 10 x^{9}}{10^{x} + x^{10}}\). Add an arbitrary constant C on each domain component.
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