\(\int \frac{10x^9+10^x \log_e 10}{x^{10}+10^x}dx\) equals
\(\int \frac{10x^9+10^x \log_e 10}{x^{10}+10^x}dx\) equals
10x - x10+C
10x+x10+C
(10x+x10)-1 +C
log(10x+x10)+C
The Correct Option is D
Solution and Explanation
Let x10+10x = t
∴ (10x9+10x loge 10) dx=dt
\(\Rightarrow \int \frac{10x^9+10^x \log_e 10}{x^{10}+10^x}dx=\int\frac{dt}{t}\)
=log t+C
=log(10x+x10)+C
Hence, the correct Answer is D.
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C