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Solve the following.
Evaluate: ∫ (10ˣ log 10 + 10x⁹)/(10ˣ + x¹⁰) dx
Evaluate:
$\int \frac{10^x \log 10 + 10x^9}{10^x + x^{10}} \, dx$
Answer
Answer
Integral: $\int \left((10^x \log(10) + 10 x^9)/(10^x + x^10)\right) dx$
The numerator is the derivative of the denominator: $\frac{d}{dx}(10^x+x^{10})=10^x\log 10+10x^9$. Put $t=10^x+x^{10}$.
$\int\frac{dt}{t}=\log|t|+C$
Final answer: $\log(10^x + x^{10}) + C$