integral of sin(2x)ln(cos(x))

Lösung

\int sin (2 x) ln (cos (x)) d x

- \frac{1}{2} ln (cos (x)) cos (2 x) - \frac{1}{2} (\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x))) + C

Schritte zur Lösung

\int sin (2 x) ln (cos (x)) d x

Wende die partielle Integration an

= - \frac{1}{2} ln (cos (x)) cos (2 x) - \int \frac{1}{2} tan (x) cos (2 x) d x

\int \frac{1}{2} tan (x) cos (2 x) d x = \frac{1}{2} (\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x)))

= - \frac{1}{2} ln (cos (x)) cos (2 x) - \frac{1}{2} (\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x)))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln (cos (x)) cos (2 x) - \frac{1}{2} (\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x))) + C

\int \frac{8 x}{4 x ^{2} - 9} d x

\int - sec^{2} (\frac{9 x}{8}) d x

\int \frac{9 ^{x}}{1 + 8 1 ^{x}} d x

\int y^{3} ln^{3} (y) d y

\int (x - 1)^{2} ln (x) d x