Question

Cho hàm số \( f(x) \) thoả mãn \( f'(x) = 4x + \frac{5}{\cos^2 x} \) và \( f\left(\frac{\pi}{4}\right) = \frac{\pi^2}{8} \). Tìm \( f(x) \)?

Answer 1

\(f'\left(x\right)=4x+\dfrac{5}{cos^2x}\)

=>\(f\left(x\right)=\int\left(4x+\dfrac{5}{cos^2x}\right)dx\)

\(=4\cdot\dfrac{1}{2}x^2+5\cdot tanx+C\)

\(=2x^2+5\cdot tanx+C\)

\(F\left(\dfrac{\Omega}{4}\right)=\dfrac{\Omega^2}{8}\)

=>\(2\cdot\left(\dfrac{\Omega}{4}\right)^2+5\cdot tan\left(\dfrac{\Omega}{4}\right)+C=\dfrac{\Omega^2}{8}\)

=>\(5\cdot1+C=0\)

=>C=-5

=>\(f\left(x\right)=2x^2+5\cdot tanx-5\)