Let I=\(∫^\frac{π}{2}_{\pi}{2}sin^7xdx.....(1)\)
\(As sin^7(−x)=(sin(−x))^7=(−sinx)^7=−sin^7x,therefore,sin^2x is an odd function.\)
\(It is known that,if f(x)is an odd function,then ∫^a_-aƒ(x)dx=0\)
\(∴I=∫^\frac{π}{2}_\frac{π}{2}sin^7xdx=0\)
If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:
The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).