$$ \left. \begin{array} { l } { \int \cos ^ { 8 } x \sin x d x } \ { \int \cos ^ { 8 } x \sin x d x = \square } \end{array} ight. $$

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$$ \left. \begin{array} { l } { \int \cos ^ { 8 } x \sin x d x } \ { \int \cos ^ { 8 } x \sin x d x = \square } \end{array} ight. $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int \cos ^ { 8 } x \sin x d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \cos^{8}\left(x\right)\sin\left(x\right) dx$$
- step1: Evaluate the integral:
  $$-\frac{\cos^{8+1}\left(x\right)}{8+1}$$
- step2: Add the numbers:
  $$-\frac{\cos^{9}\left(x\right)}{8+1}$$
- step3: Add the numbers:
  $$-\frac{\cos^{9}\left(x\right)}{9}$$
- step4: Simplify the expression:
  $$-\frac{1}{9}\cos^{9}\left(x\right)$$
- step5: Add the constant of integral C:
  $$-\frac{1}{9}\cos^{9}\left(x\right) + C, C \in \mathbb{R}$$
The integral of $$ \int \cos ^ { 8 } x \sin x d x $$ is equal to $$ -\frac{1}{9}\cos^{9}(x) + C $$, where $$ C $$ is an arbitrary constant.
$$ \int \cos ^ { 8 } x \sin x d x = -\frac{1}{9}\cos^{9}(x) + C $$

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