Let \(∫^\frac{π}{2}_\frac{-π}{2}sin^2xdx\)
\(As\, sin^2(−x)=(sin(−x))^2=(−sinx)^2=sin^2x\,therefore,\,sin^2x\,\, is\,\, an\,\, even\,\, function.\)
\(It\,\, is\,\, known\,\, that\,\, if\,\, f(x)\,is\,\, an\,\, even\, function,then\,\, ∫^a_{-a}ƒ(x)dx=2∫^a_0ƒ(x)dx\)
\(I=2∫_0^{π}{2} sin^2xdx\)
\(=2∫_0^{π}{2} \frac{1-cos2x}{2}dx\)
\(=∫_0^\frac{π}{2}(1-cos2x)dx\)
\(=[x-\frac{sin2x}{2}]^\frac{π}{2}_0\)
\(=\frac{π}{2}\)
If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:
The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).