What is the integral of x/(x^2 – a^2)
The integral of
\frac{x}{x^2 - a^2} is
\frac{1}{2}\ln \lvert x^2 - a^2 \rvert + C.
Solution of the integral of x/(x^2 – a^2)
Solution: Before we determine the integral of
\frac{x}{x^2 - a^2}, lets us recall what we need to show exactly:
\begin{equation*}
\int \frac{x}{x^2 - a^2} dx.
\end{equation*}
We use the substitution method where
u = x^2 - a^2 such that we get the derivative
du = 2xdx \iff xdx = \frac{1}{2}du. Now we will implement that in the integral above, and therefore, we get the desired solution:
\begin{align*}
\int \frac{x}{x^2 - a^2} dx &= \int \frac{\frac{1}{2}du}{u} \\
&= \frac{1}{2} \int \frac{du}{u} \\
&= \frac{1}{2} \ln \lvert u \rvert + C \\
&= \frac{1}{2} \ln \lvert x^2 - a^2 \rvert + C
\end{align*}Conclusion
The detailed solution above shows us that the integral of
\frac{x}{x^2 - a^2} is
\frac{1}{2}\ln \lvert x^2 - a^2 \rvert + C.
