integral of cos(2θ)(sin(θ)+cos(θ))^2

Lösung

\int cos (2 θ) (sin (θ) + cos (θ))^{2} d θ

\frac{1}{4} (sin (θ) + cos (θ))^{4} + C

Schritte zur Lösung

\int cos (2 θ) (sin (θ) + cos (θ))^{2} d θ

Wende U-Substitution an

= \int \frac{u}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int u d u

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{u ^{2}}{2}

Setze in

u = (sin (θ) + cos (θ))^{2}

ein

= \frac{1}{2} \cdot \frac{( ( sin ( θ ) + cos ( θ ) ) ^{2} ) ^{2}}{2}

Vereinfache

\frac{1}{2} \cdot \frac{( ( sin ( θ ) + cos ( θ ) ) ^{2} ) ^{2}}{2} : \frac{1}{4} (sin (θ) + cos (θ))^{4}

= \frac{1}{4} (sin (θ) + cos (θ))^{4}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} (sin (θ) + cos (θ))^{4} + C

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