\(\mathrm{D}\sin x = \cos x\)
Kertauksena: Sinin ja kosinin derivaatta
\(\mathrm{D}\cos x = -\sin x\)
Derivointikaavat:
\(\int{\sin x\ \text{d}x}=-\cos x+C\)
\(\int{\cos x\ \text{d}x}=\sin x+C\)
Integrointikaavat:
Esim. \(\int 3 \cos x + 5 \sin x \, \text{d}x\)
\(=3\sin x + 5 \cdot(-\cos x) + C\)
\(=3\sin x - 5 \cos x + C\)
\(\int{{f}'(x)\ {g}'(f(x))\ \text{d}x}=g(f(x))+C\)
\(\text{D}g(f(x))=g'(f(x))f'(x)\)
Esim. \(\int \cos (5x)\, \text{d} x\)
\(=\dfrac{1}{5}\int 5 \cos (5x)\, \text{d} x\)
\(=\dfrac{1}{5}\sin (5x)+C\).
\(f(x)=5x\)
\(f'(x)=5\)
\(g'(x)=\cos x\)
\(g(x)=\sin x\)
\(=\int \dfrac{1}{5}\cdot 5 \cos (5x)\, \text{d} x\)
Huom! Mitään x:ää sisältävää ei saa siirtää integraalin ulkopuolelle!
\(f(x)=4x^3\)
\(f'(x)=12x^2\)
\(g'(x)=\sin x\)
\(g(x)=-\cos x\)
Esim. \(\int 3x^2 \sin (4x^3)\, \text{d} x\)
\(=\dfrac{1}{4}\int 12x^2 \sin (4x^3) \,\text{d}x\)
\(=-\dfrac{1}{4}\cos (4x^3) + C\)
By Timo Pelkola
