The value of $\int \frac{10^{x/2}}{\sqrt{10^{-x} - 10^{x}}}dx$ is
- $\frac{1}{\log_{e} 10} \sin^{-1} \left(10^{x} \right)+c $
- $2\sqrt{10^{-x} + 10^{x} } +c $
- $\frac{1}{\log_{e} 10}\sinh^{-1} \left(10x\right)+c$
- $\frac{-1}{\log_{e} 10}\sinh^{-1} \left(10x\right)+c$
The Correct Option is A
Solution and Explanation
$= \int \frac{10^{\frac{x}{2}} .10^{\frac{x}{2}}}{\sqrt{1-\left(10^{x}\right)^{2}}} dx = \int\frac{10^{x} }{\sqrt{1-\left(10^{x}\right)^{2}}}dx $
Put $10^{x} =t \Rightarrow 10^{x} \log10 dx =dt $
$\therefore \, \, I = \frac{1}{\log 10} \int \frac{dt}{\sqrt{1-t^{2}} } = \frac{1}{\log 10}\left(\sin^{-1} t\right)+c $
$= \frac{1}{\log 10}\sin^{-1} \left(10^{x}\right)+c $
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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
Integration Using Trigonometric Identities
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