NIMCET 2010 — Mathematics PYQ

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Question

NIMCET 2010 — Mathematics PYQ

NIMCET | Mathematics | 2010

$\int \log_{10} x \, dx$ is:

Choose the correct answer:

Explanation

Solution

1. Change of Base Rule

The given integral is in base 10. To integrate, it is easier to convert it to natural log (base $e$ ) using the change of base formula:

\log_{10} x = \frac{\log_e x}{\log_e 10} = \log_{10} e \cdot \log_e x

Now, substitute this into the integral:

I = \int \log_{10} e \cdot \log_e x \, dx

I = \log_{10} e \int \log_e x \, dx

2. Integrate $\log_e x$

Using integration by parts ( $\int u \, dv = uv - \int v \, du$ ) where $u = \log_e x$ and $dv = dx$ :

\int \log_e x \, dx = x \log_e x - x + c

Factor out $x$ :

\int \log_e x \, dx = x(\log_e x - 1) + c

Since $1 = \log_e e$ , we can use the property $\log a - \log b = \log(\frac{a}{b})$ :

\int \log_e x \, dx = x(\log_e x - \log_e e) + c

\int \log_e x \, dx = x \log_e \left( \frac{x}{e} \right) + c

3. Combine the Steps

Multiply the result by the constant $\log_{10} e$ :

I = \log_{10} e \cdot x \log_e \left( \frac{x}{e} \right) + c

Correct Option:

The correct option is (c) $\log_{10} e \cdot x \log_e \left( \frac{x}{e} \right) + c$ .