Aloha :)
$$I=\int4\cos^2\frac x2\,dx$$$$\phantom I=\int\underbrace{4\cos\frac x2}_{=u}\cdot\underbrace{\cos\frac x2}_{=v'}\,dx=\underbrace{4\cos\frac x2}_{=u}\cdot\underbrace{2\sin\frac x2}_{=v}-\int\underbrace{\left(-2\sin\frac x2\right)}_{=u'}\cdot\underbrace{2\sin\frac x2}_{=v}\,dx$$$$\phantom I=\underbrace{8\cos\frac x2\sin\frac x2}_{=4\sin x}+\int4\sin^2\frac x2\,dx=4\sin x+\int4\left(1-\cos^2\frac x2\right)dx$$$$\phantom I=4\sin x+4x-\int4\cos^2\frac x2\,dx=4\sin x+4x-I$$Diese Gleichung kannst du nach \(I\) auflösen:$$I=\int4\cos^2\frac x2\,dx=2x+2\sin x+\text{const}$$