\(I=\)\(∫_0^{\frac \pi2}cos^2x\ dx\) ...(1)
⇒ \(I\) =\(∫_0^{\frac \pi2}cos^2(\frac \pi2-x)\ dx\) \((∫_0^aƒ(x)dx = ƒ(a-x)dx)\)
⇒\(I\) = \(∫_0^{\frac \pi2}sin^2x\ dx\) ...(2)
Adding (1) and (2), we obtain
\(2I\) =\(∫_0^{\frac \pi2}(sin ^2x+cos^2x)\ dx\)
⇒\(2I\) = \(∫_0^{\frac \pi2}1\ dx\)
⇒\(2I\) = \([x]_0^{\frac \pi2}\)
⇒\(2I\) = \(\frac \pi2\)
⇒\(I\) = \(\frac \pi4\)
If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:
The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).