Question

\(\int\frac{4x^9}{x^{10}-10}dx=\)

• A. \(\frac{1}{5}\log|x^{10}-10|+C\)
• B. \(\frac{2}{5}\log|x^{10}-10|+C\)
• C. \(\frac{1}{10}\log|x^{10}-10|+C\)
• D. \(\frac{-2}{5}\log|x^{10}-10|+C\)
• E. \(\frac{-1}{10}\log|x^{10}-10|+C\)

Answer 1

The Correct Option is B

We are given the integral \( \int \frac{4x^9}{x^{10} - 10} \, dx \) and are asked to find the solution.

We can solve this by using substitution. Let: 

\( u = x^{10} - 10 \),

so that \( du = 10x^9 \, dx \), or equivalently \( \frac{du}{10} = x^9 \, dx \).

Now, rewrite the integral in terms of \( u \):

\( \int \frac{4x^9}{x^{10} - 10} \, dx = \frac{4}{10} \int \frac{1}{u} \, du = \frac{2}{5} \int \frac{1}{u} \, du \).

The integral of \( \frac{1}{u} \) is \( \log|u| \), so we have:

\( \frac{2}{5} \log|u| + C \).

Substitute back \( u = x^{10} - 10 \):

\( \frac{2}{5} \log|x^{10} - 10| + C \).

The correct answer is \( \frac{2}{5} \log|x^{10} - 10| + C \).