„Integrálás - Alapvető integrálok” változatai közötti eltérés

#:a) $$\int \left(3+4x+5x^{2}\right)dx$$  

#:b) $$\int_{-1}^{2}\left(x^{2}-\sin(5x)\right)dx$$  

#:c) $$\int_{0}^{\pi}\sin^{2}x\,dx$$  

#:d) $$\int_{0}^{\pi}\cos^{2}x\,dx$$  

#:e) $$\int\sqrt{3x+2}dx$$  

#:f) $$\int\frac{1}{2x}dx$$</wlatex><includeonly></includeonly><noinclude>

<wlatex>#:a)$$\int \left(3+4x+5x^{2}\right)dx=3x+2x^{2}+\frac{5}{3}x^{3}+C$$  

#:b)$$\int_{-1}^{2}\left(x^{2}-\sin(5x)\right)dx=\left[\frac{x^{3}}{3}+\frac{\cos(5x)}{5}\right]^{2}_{-1}=\frac{8}{3}+\frac{\cos 10}{5}-\left(-\frac{1}{3}+\frac{\cos 5}{5}\right)\approx 2.78$$  

#:c)$$\int_{0}^{\pi}\sin^{2}x\,dx=\int_{0}^{\pi}\left(\frac{1}{2}-\frac{\cos(2x)}{2}\right)dx=\left[\frac{x}{2}\right]^{\pi}_{0}-\left[\frac{\sin(2x)}{4}\right]^{\pi}_{0}=\frac{\pi}{2}$$  

#:d)$$\int_{0}^{\pi}\cos^{2}x\,dx=\int_{0}^{\pi}\left(\frac{1}{2}+\frac{\cos(2x)}{2}\right)dx=\frac{\pi}{2}$$  

#:e)$$\int\sqrt{3x+2}dx=\int \left(3x+2\right)^{\frac{1}{2}}dx=2(3x+2)^{\frac{3}{2}}+C$$  

#:f)$$\int\frac{1}{2x}dx=\frac{\ln (2x)}{2}+C$$ vagy $$\int\frac{1}{2x}dx=\frac{1}{2}\int\frac{1}{x}dx=\frac{1}{2}\ln x+C'$$ $$C'=C+\frac{\ln 2}{2}$$</wlatex>