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Exercise

$\int\frac{\cos^3\left(x\right)}{\sec^4\left(x\right)}dx$

Step-by-step Solution

1

Riscrivere l'espressione trigonometrica $\frac{\cos\left(x\right)^3}{\sec\left(x\right)^4}$ all'interno dell'integrale

$\int\cos\left(x\right)^{7}dx$
2

Applicare la formula: $\int\cos\left(\theta \right)^ndx$$=\frac{\cos\left(\theta \right)^{\left(n-1\right)}\sin\left(\theta \right)}{n}+\frac{n-1}{n}\int\cos\left(\theta \right)^{\left(n-2\right)}dx$, dove $n=7$

$\frac{\cos\left(x\right)^{6}\sin\left(x\right)}{7}+\frac{6}{7}\int\cos\left(x\right)^{5}dx$
3

L'integrale $\frac{6}{7}\int\cos\left(x\right)^{5}dx$ risulta in: $\frac{6\cos\left(x\right)^{4}\sin\left(x\right)}{35}+\frac{8}{35}\cos\left(x\right)^{2}\sin\left(x\right)+\frac{16}{35}\sin\left(x\right)$

$\frac{6\cos\left(x\right)^{4}\sin\left(x\right)}{35}+\frac{8}{35}\cos\left(x\right)^{2}\sin\left(x\right)+\frac{16}{35}\sin\left(x\right)$
4

Raccogliere i risultati di tutti gli integrali

$\frac{\cos\left(x\right)^{6}\sin\left(x\right)}{7}+\frac{16}{35}\sin\left(x\right)+\frac{8}{35}\cos\left(x\right)^{2}\sin\left(x\right)+\frac{6\cos\left(x\right)^{4}\sin\left(x\right)}{35}$
5

Poiché l'integrale che stiamo risolvendo è un integrale indefinito, quando finiamo di integrare dobbiamo aggiungere la costante di integrazione $C$

$\frac{\cos\left(x\right)^{6}\sin\left(x\right)}{7}+\frac{16}{35}\sin\left(x\right)+\frac{8}{35}\cos\left(x\right)^{2}\sin\left(x\right)+\frac{6\cos\left(x\right)^{4}\sin\left(x\right)}{35}+C_0$

Final answer to the exercise

$\frac{\cos\left(x\right)^{6}\sin\left(x\right)}{7}+\frac{16}{35}\sin\left(x\right)+\frac{8}{35}\cos\left(x\right)^{2}\sin\left(x\right)+\frac{6\cos\left(x\right)^{4}\sin\left(x\right)}{35}+C_0$

Try other ways to solve this exercise

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  • Integrate by partial fractions
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  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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