Hi i have one math question that i am struggling on.
which was just simple integral question range from 0 to 24
and so i applied integral of arctan x property here as above(Blue).
but answer is the 124.102 which is about 5times of my answer.
i guess i made mistakes substituting x/5 into integral property.
But this seems correct way to substitute. Hope somebody help me out for this.
Best Answer
\begin{equation} \begin{aligned} \int_0^{24} 5 \tan ^{-1}\left(\frac{x}{5}\right) d x=&\left[5 x \tan ^{-1}\left(\frac{x}{5}\right)\right]_0^{24} - 5 \int_0^{24} \frac{x}{1+\left(\frac{x}{5}\right)^2} \cdot \frac{1}{5} d x \\ = & 120 \tan ^{-1}\left(\frac{24}{5}\right)-25 \int_0^{24} \frac{x}{25+x^2} d x \\ = & 120 \tan ^{-1}\left(\frac{24}{5}\right)-\frac{25}{2}\left[\ln \left(25+x^2\right)\right]_0^{24} \\ = & 120 \tan ^{-1}\left(\frac{24}{5}\right)-\frac{25}{2} \ln \left(\frac{576}{25}\right) \end{aligned} \end{equation}