Question

tìm nguyên hàm F(x) của hàm số f(x) thỏa mãn điều kiện tương ứng

a) \(f\left(x\right)3-5sinx\) và \(F\left(\dfrac{\pi}{2}\right)=2\)

b) \(f\left(x\right)=x-2cosx\) và \(F\left(\pi\right)=\dfrac{1}{2}\)

Answer 1

a: f(x)=3-5sinx

=>\(F\left(x\right)=\int\left(-5\cdot sinx+3\right)dx=5\cdot cosx+3x+C\)

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

=>\(5\cdot cos\left(\dfrac{\Omega}{2}\right)+\dfrac{3\Omega}{2}+C=2\)

=>\(C=2-\dfrac{3}{2}\Omega\)

=>\(F\left(x\right)=5cosx+3x+2-\dfrac{3}{2}\Omega\)

b: f(x)=x-2*cosx

=>\(F\left(x\right)=\int\left(x-2\cdot cosx\right)dx=\dfrac{1}{2}x^2-2\cdot sinx+C\)

\(F\left(\Omega\right)=\dfrac{1}{2}\)

=>\(\dfrac{1}{2}\cdot\Omega^2-2\cdot sin\left(\Omega\right)+C=\dfrac{1}{2}\)

=>\(C=\dfrac{1}{2}-\dfrac{1}{2}\Omega^2\)

=>\(F\left(x\right)=\dfrac{1}{2}x^2-2\cdot sinx+\dfrac{1}{2}-\dfrac{1}{2}\Omega^2\)