de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (3y)/(sqrt(2y^2+5y))

Lösung

\int \frac{3 y}{2 y ^{2} + 5 y} d y

Lösung

\frac{3}{4 2} 16 y^{2} + 40 y - 5 ln \frac{16 y ^{2} + 40 y + 4 y + 5}{5} + C

Schritte zur Lösung

\int \frac{3 y}{2 y ^{2} + 5 y} d y

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{y}{2 y ^{2} + 5 y} d y

Vervollständige das Quadrat

2 y^{2} + 5 y : 2 (y + \frac{5}{4})^{2} - \frac{25}{8}

= 3 \cdot \int \frac{y}{2 ( y + \frac{5}{4} ) ^{2} - \frac{25}{8}} d y

Wende U-Substitution an

= 3 \cdot \int \frac{4 u - 5}{2 16 u ^{2} - 25} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \frac{1}{2} \cdot \int \frac{4 u - 5}{16 u ^{2} - 25} d u

Multipliziere aus

\frac{4 u - 5}{16 u ^{2} - 25} : \frac{4 u}{16 u ^{2} - 25} - \frac{5}{16 u ^{2} - 25}

= 3 \cdot \frac{1}{2} \cdot \int \frac{4 u}{16 u ^{2} - 25} - \frac{5}{16 u ^{2} - 25} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 \cdot \frac{1}{2} (\int \frac{4 u}{16 u ^{2} - 25} d u - \int \frac{5}{16 u ^{2} - 25} d u)

\int \frac{4 u}{16 u ^{2} - 25} d u = \frac{1}{4} 16 u^{2} - 25

\int \frac{5}{16 u ^{2} - 25} d u = \frac{5}{4} ln (\frac{16 u ^{2} - 25 + 4 u}{5})

= 3 \cdot \frac{1}{2} (\frac{1}{4} 16 u^{2} - 25 - \frac{5}{4} ln (\frac{16 u ^{2} - 25 + 4 u}{5}))

Setze in

u = y + \frac{5}{4}

ein

= 3 \cdot \frac{1}{2} \frac{1}{4} 16 (y + \frac{5}{4})^{2} - 25 - \frac{5}{4} ln \frac{16 ( y + \frac{5}{4} ) ^{2} - 25 + 4 ( y + \frac{5}{4} )}{5}

Vereinfache

3 \cdot \frac{1}{2} \frac{1}{4} 16 (y + \frac{5}{4})^{2} - 25 - \frac{5}{4} ln \frac{16 ( y + \frac{5}{4} ) ^{2} - 25 + 4 ( y + \frac{5}{4} )}{5} : \frac{3}{4 2} 16 y^{2} + 40 y - 5 ln \frac{16 y ^{2} + 40 y + 4 y + 5}{5}

= \frac{3}{4 2} 16 y^{2} + 40 y - 5 ln \frac{16 y ^{2} + 40 y + 4 y + 5}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{4 2} 16 y^{2} + 40 y - 5 ln \frac{16 y ^{2} + 40 y + 4 y + 5}{5} + C

Graph

Beliebte Beispiele

integral of (x^3-2x+1)sin(4x)

\int (x^{3} - 2 x + 1) sin (4 x) d x

integral of (6-z^2)/(2z^2+z-21)

\int \frac{6 - z ^{2}}{2 z ^{2} + z - 21} d z

integral of sin(3x)e^{3x+1}

\int sin (3 x) e^{3 x + 1} d x

integral of 3pi

\int 3 π d x

integral of (3x^4y^2)/(sqrt(9y^2-4x^2))

\int \frac{3 x ^{4} y ^{2}}{9 y ^{2} - 4 x ^{2}} d y