Question:
$\int\frac{dx}{x-\sqrt{x}}$ is equal to
$\int\frac{dx}{x-\sqrt{x}}$ is equal to
Updated On: Apr 8, 2024
- $2\, log\, \left|\sqrt{x}-1\right|+C$
- $2\, log\, \left|\sqrt{x}+1\right|+C$
- $log\, \left|\sqrt{x}-1\right|+C$
- $\frac{1}{2}log\, \left|\sqrt{x}+1\right|+C$
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The Correct Option is A
Solution and Explanation
Let $I=\int \frac{d x}{x-\sqrt{x}}=\int \frac{d x}{\sqrt{x}(\sqrt{x}-1)}$
Put $\sqrt{x}-1=t$
$\Rightarrow \frac{1}{2 \sqrt{x}} d x=d t$
$\therefore I=\int \frac{2 d t}{t}=2 \log |t|+C$
$=2 \log |\sqrt{x}-1|+C$
Put $\sqrt{x}-1=t$
$\Rightarrow \frac{1}{2 \sqrt{x}} d x=d t$
$\therefore I=\int \frac{2 d t}{t}=2 \log |t|+C$
$=2 \log |\sqrt{x}-1|+C$
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
