a: \(f\left(x\right)=cos^2x=\frac{1+cos2x}{2}\)
=>\(F\left(x\right)=\int\frac{1+cos2x}{2}\cdot dx=\frac12\cdot\int\left(1+cos2x\right)\cdot\left(dx\right)=\frac12\cdot\left\lbrack x+\frac12\cdot\sin2x+C\right\rbrack\)
\(F\left(\frac{\pi}{2}\right)=1\)
=>\(\frac12\cdot\left\lbrack\frac{\pi}{2}+\frac12\cdot\sin\left(\pi\right)+C\right\rbrack=1\)
=>\(\frac12\left(\frac{\pi}{2}+C\right)=1\)
=>\(C+\frac{\pi}{2}=2\)
=>\(C=2-\frac{\pi}{2}\)
=>\(F\left(x\right)=\frac12x+\frac14\cdot\sin2x+2-\frac{\pi}{2}\)
b: \(f\left(x\right)=\sin^2x=\frac{1-cos2x}{2}\)
=>\(F\left(x\right)=\int\frac{1-cos2x}{2}\cdot dx=\frac12\cdot\int\left(1-cos2x\right)\cdot\left(dx\right)=\frac12\cdot\left\lbrack x-\frac12\cdot\sin2x+C\right\rbrack\)
\(F\left(\frac{\pi}{4}\right)=10\)
=>\(\frac12\left\lbrack\frac{\pi}{4}-\frac12\cdot\sin\left(\frac{\pi}{2}\right)+C\right\rbrack=10\)
=>\(\frac{\pi}{4}-\frac12+C=20\)
=>\(C=20+\frac12-\frac{\pi}{4}=20,5-\frac{\pi}{4}\)
=>\(F\left(x\right)=\frac12x-\frac14\cdot\sin2x+20,5-\frac{\pi}{4}\)
