^{Why do we write the integration of : dx/x as
log|x| + log|C| in some questions and log|x| + C in other questions? (While solving differential equations)}

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Solution

If we want the constant of integration to be submerged in log term itself we write logC and in otherwise cases we will go for C

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In some of the cases we can split the integrand into the sum of the two functions such that the integration of one of them by parts produces an integral which cancels the other integral. Suppose we have an integral of the type
$\int [f (x) h (x) + g (x)] d x$
Let $\int f (x) h (x) d x = I_{1}$ and $\int g (x) d x = I_{2}$
Integrating $I_{1}$ by parts, we get
$I_{1} = f (x) \int h (x) d x - \int {f^{'} (x) \int h (x) d x} d x$
Now find $\int (\frac{1}{l o g x} - \frac{1}{(l o g x)^{2}}) d x$ (x > 0)

Q. Solve differential equation :

x log x \frac{d y}{d x} + y = 2 log x

The general solution of differential equation $\frac{d y}{d x} = l o g x$ is

Q. The integrating factor of the differential equation (x log x)

\frac{d y}{d x} + y = 2 \log x

, is given by
(a) log (log x)

(b) e^x

(c) log x

(d) x

Q. The integrating factor of the differential equation

x \frac{d y}{d x} + 2 y = x^{2}

(x \neq 0)

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Why do we write the integration of : dx/x as log|x| + log|C| in some questions and log|x| + C in other questions? (While solving differential equations)

If we want the constant of integration to be submerged in log term itself we write logC and in otherwise cases we will go for C

^{Why do we write the integration of : dx/x as
log|x| + log|C| in some questions and log|x| + C in other questions? (While solving differential equations)}