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Beliebt Rechnen >

integral of sin(x)(sec^2(x)+cot^2(x))

Lösung

\int sin (x) (sec^{2} (x) + cot^{2} (x)) d x

Lösung

sec (x) + ln tan (\frac{x}{2}) + cos (x) + C

Schritte zur Lösung

\int sin (x) (sec^{2} (x) + cot^{2} (x)) d x

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{sin ^{2} ( x ) + cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )} d x

Multipliziere aus

\frac{sin ^{2} ( x ) + cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )} : \frac{sin ( x )}{cos ^{2} ( x )} + \frac{cos ^{2} ( x )}{sin ( x )}

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

Wende die Summenregel an:

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

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

\int \frac{sin ( x )}{cos ^{2} ( x )} d x = sec (x)

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

= sec (x) + ln tan (\frac{x}{2}) + cos (x)

Füge eine Konstante zur Lösung hinzu

= sec (x) + ln tan (\frac{x}{2}) + cos (x) + C

Graph

Beliebte Beispiele

integral of (x^2+2)/(x^2-4)

\int \frac{x ^{2} + 2}{x ^{2} - 4} d x

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

\int \frac{1}{x} \frac{1 + x}{x} d x

integral of (x+1)/(x^2-5x+6)

\int \frac{x + 1}{x ^{2} - 5 x + 6} d x

integral of (3/(1+x^2)-2/(sqrt(1-x^2)))

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

integral of (x+3)/(x^2-5x+6)

\int \frac{x + 3}{x ^{2} - 5 x + 6} d x