integral from x to 2x of ln(x)cos(3x)

Lösung

\int_{x}^{2 x} ln (x) cos (3 x) d x

\frac{1}{3} (ln (2 x) sin (6 x) - Si (3 \cdot 2 x)) - \frac{1}{3} (ln (x) sin (3 x) - Si (3 x))

Schritte zur Lösung

\int_{x}^{2 x} ln (x) cos (3 x) d x

Wende die partielle Integration an

= [\frac{1}{3} ln (x) sin (3 x) - \int \frac{sin ( 3 x )}{3 x} d x]_{x}^{2 x}

\int \frac{sin ( 3 x )}{3 x} d x = \frac{1}{3} Si (3 x)

= [\frac{1}{3} ln (x) sin (3 x) - \frac{1}{3} Si (3 x)]_{x}^{2 x}

Vereinfache

[\frac{1}{3} ln (x) sin (3 x) - \frac{1}{3} Si (3 x)]_{x}^{2 x} : [\frac{1}{3} (ln (x) sin (3 x) - Si (3 x))]_{x}^{2 x}

= [\frac{1}{3} (ln (x) sin (3 x) - Si (3 x))]_{x}^{2 x}

Berechne die Grenzen:

\frac{1}{3} (ln (2 x) sin (6 x) - Si (3 \cdot 2 x)) - \frac{1}{3} (ln (x) sin (3 x) - Si (3 x))

= \frac{1}{3} (ln (2 x) sin (6 x) - Si (3 \cdot 2 x)) - \frac{1}{3} (ln (x) sin (3 x) - Si (3 x))

\int_{0}^{2} (2 x + 3) d x

\int_{0}^{2} 5 x^{2} + 2 d x

\int_{- 3}^{3} \frac{1}{( x - 1 ) ^{4}} d x

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\int_{- 3}^{3} (- 5 x^{2} - 6 x)^{3} d x