Question:
The value of
\[
\int \frac{1}{x^1} \, dx, { is } \left[ \frac{(x - 1)^3}{(x + 2)^5} \right]_1^4
\]
The value of
\[
\int \frac{1}{x^1} \, dx, { is } \left[ \frac{(x - 1)^3}{(x + 2)^5} \right]_1^4
\]
Show Hint
When solving definite integrals involving rational functions, make sure to simplify the expression and evaluate the limits correctly.
Updated On: Mar 26, 2025
- \( \frac{4}{3} \frac{x + 1}{x - 2} \left( \frac{1}{4} \right) + C \)
- \( \frac{3}{4} \frac{x - 1}{x + 2} \left( \frac{1}{4} \right) + C \)
- \( \frac{4}{3} \frac{x - 1}{x + 2} \left( \frac{1}{4} \right) + C \)
- \( \frac{1}{3} \frac{2x - 1}{4x - 3} \left( \frac{1}{4} \right) + C \)
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The Correct Option is C
Solution and Explanation
We are asked to find the value of the following definite integral: \[ \int \frac{1}{x^1} \, dx \] We are given the expression for the result in the form of a complex rational function. By solving the integral: \[ \int \frac{1}{x^1} \, dx = \frac{(x - 1)^3}{(x + 2)^5} \left[ \right]_1^4 \] This simplifies to the expression as seen in Option C.
Therefore, the correct answer is Option C.
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