integral of ln(sqrt(x)+sqrt(1+x))

Lösung

\int ln (x + 1 + x) d x

x ln (x + 1 + x) - x x + 1 + \frac{1}{2} \frac{x + 1}{x} x + \frac{1}{2} ln x + 1 + x + C

Schritte zur Lösung

\int ln (x + 1 + x) d x

Wende die partielle Integration an

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

\int \frac{x}{2 x + 1} d x = x x + 1 - \frac{1}{2} \frac{x + 1}{x} x - \frac{1}{2} ln x + 1 + x

= x ln (x + 1 + x) - (x x + 1 - \frac{1}{2} \frac{x + 1}{x} x - \frac{1}{2} ln x + 1 + x)

Vereinfache

= x ln (x + 1 + x) - x x + 1 + \frac{1}{2} \frac{x + 1}{x} x + \frac{1}{2} ln x + 1 + x

Füge eine Konstante zur Lösung hinzu

= x ln (x + 1 + x) - x x + 1 + \frac{1}{2} \frac{x + 1}{x} x + \frac{1}{2} ln x + 1 + x + C

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

d er i v a t i v e f (x) = \frac{3 x - 7}{5 x - 2}

a re a y = 20 - x, y = x, y = 1

y^{^{''}} + 19 y = 0

\int_{0}^{\frac{π}{2}} tan^{2} (x) d x