2021-12-29

Trigonometric integrals. Evaluate the following integrals.
$\int {\mathrm{sin}}^{3}x{\mathrm{cos}}^{5}xdx$

Wendy Boykin

Step 1
Given: The integral $\int {\mathrm{sin}}^{3}x{\mathrm{cos}}^{5}xdx$
To evaluate: the integration of given trigonometric integral.
Step 2
Explanation:
Let $I=\int {\mathrm{sin}}^{3}x{\mathrm{cos}}^{5}xdx$
This can be re-written as
$I=\int \mathrm{sin}x{\mathrm{sin}}^{2}x{\mathrm{cos}}^{5}xdx$

$⇒I=\int \mathrm{sin}x\left({\mathrm{cos}}^{5}x-{\mathrm{cos}}^{7}x\right)dx$
Now substituting $\mathrm{cos}x=t$
$⇒-\mathrm{sin}xdx=dt$
$⇒\mathrm{sin}xdx=-dt$
Hence
$I=\int \left({t}^{5}-{t}^{7}\right)\left(-dt\right)$
$⇒I=-\int \left({t}^{5}-{t}^{7}\right)dt$
$⇒I=-\left(\frac{{t}^{6}}{6}-\frac{{t}^{8}}{8}\right)+C$
$⇒I=\frac{{t}^{8}}{8}-\frac{{t}^{6}}{6}+C$
Now as $t=\mathrm{cos}x$
Therefore
$I=\frac{{\mathrm{cos}}^{8}x}{8}-\frac{{\mathrm{cos}}^{6}x}{6}+C$ where C is arbitrary constant and also known as "constant of integration"
Answer: $\int {\mathrm{sin}}^{3}x{\mathrm{cos}}^{5}xdx=\frac{{\mathrm{cos}}^{8}x}{8}-\frac{{\mathrm{cos}}^{6}x}{6}+C$ where C is constant of integration.

Bob Huerta

$\int {\mathrm{cos}}^{5}\left(x\right){\mathrm{sin}}^{3}\left(x\right)dx$
$=\int -{\mathrm{cos}}^{5}\left(x\right)\left({\mathrm{cos}}^{2}\left(x\right)-1\right)\cdot \mathrm{sin}\left(x\right)dx$
$=\int {u}^{5}\left({u}^{2}-1\right)du$
$=\int \left({u}^{7}-{u}^{5}\right)du$
$=\int {u}^{7}du-\int {u}^{5}du$
$\int {u}^{7}du$
$=\frac{{u}^{8}}{8}$
$\int {u}^{5}du$
$=\frac{{u}^{6}}{6}$
$\int {u}^{7}du-\int {u}^{5}du$
$=\frac{{u}^{8}}{8}-\frac{{u}^{6}}{6}$
$=\frac{{\mathrm{cos}}^{8}\left(x\right)}{8}-\frac{{\mathrm{cos}}^{6}\left(x\right)}{6}$
$\int {\mathrm{cos}}^{5}\left(x\right){\mathrm{sin}}^{3}\left(x\right)dx$
$=\frac{{\mathrm{cos}}^{8}\left(x\right)}{8}-\frac{{\mathrm{cos}}^{6}\left(x\right)}{6}+C$

karton

$\begin{array}{}\int \mathrm{sin}\left(x{\right)}^{3}\mathrm{cos}\left(x{\right)}^{5}dx\\ \int {t}^{3}-2{t}^{5}+{t}^{7}dt\\ \int {t}^{3}dt-\int 2{t}^{5}dt+\int {t}^{7}dt\\ \frac{{t}^{4}}{4}-\frac{{t}^{6}}{3}+\frac{{t}^{8}}{8}\\ \frac{\mathrm{sin}\left(x{\right)}^{4}}{4}-\frac{\mathrm{sin}\left(x{\right)}^{6}}{3}+\frac{\mathrm{sin}\left(x{\right)}^{8}}{8}\\ \frac{\mathrm{sin}\left(x{\right)}^{4}}{4}-\frac{\mathrm{sin}\left(x{\right)}^{6}}{3}+\frac{\mathrm{sin}\left(x{\right)}^{8}}{8}+C\end{array}$