3.2 Trigonometric integrals  (Page 4/6)

${\text{sin}}^{2}x+\_\_\_\_\_\_\_=1$

${\text{sec}}^{2}x-1=\_\_\_\_\_\_\_$

${\text{sin}}^{2}x=\_\_\_\_\_\_\_$

${\text{cos}}^{2}x=\_\_\_\_\_\_\_$

${\displaystyle \int {\text{sin}}^{3}x\text{cos}xdx}$

${\displaystyle \int \sqrt{\text{cos}x}\text{sin}xdx}$

${\displaystyle \int {\text{tan}}^5(2x){\text{sec}}^2(2x)dx}$

${\displaystyle \int {\text{sin}}^7(2x)\text{cos}(2x)dx}$

${\displaystyle \int \text{tan}\left(\frac{x}{2}\right){\text{sec}}^2\left(\frac{x}{2}\right)dx}$

${\displaystyle \int {\text{tan}}^{2}x{\text{sec}}^{2}xdx}$

${\displaystyle \int {\text{sin}}^{3}xdx}$

${\displaystyle \int {\text{cos}}^{3}xdx}$

${\displaystyle \int \text{sin}x\text{cos}xdx}$

${\displaystyle \int {\text{cos}}^{5}xdx}$

${\displaystyle \int {\text{sin}}^{5}x{\text{cos}}^{2}xdx}$

${\displaystyle \int {\text{sin}}^{3}x{\text{cos}}^{3}xdx}$

${\displaystyle \int \sqrt{\text{sin}x}}\text{cos}xdx$

${\displaystyle \int \sqrt{\text{sin}x}}{\text{cos}}^{3}xdx$

${\displaystyle \int \text{sec}x\text{tan}xdx}$

${\displaystyle \int \text{tan}(5x)dx}$

${\displaystyle \int {\text{tan}}^{2}x\text{sec}xdx}$

${\displaystyle \int \text{tan}x{\text{sec}}^{3}xdx}$

${\displaystyle \int {\text{sec}}^{4}xdx}$

${\displaystyle \int \text{cot}xdx}$

${\displaystyle \int \text{csc}xdx}$

${\displaystyle \int \frac{{\text{tan}}^{3}x}{\sqrt{\text{sec}x}}dx}$

${\displaystyle \int {\text{sin}}^{2}ax\text{cos}axdx}$

${\displaystyle \int \text{sin}ax\text{cos}axdx.}$

${\displaystyle {\int }_0^{\pi }{\text{sin}}^{2}xdx}$

${\displaystyle {\int }_0^{\pi }{\text{sin}}^{4}xdx}$

${\displaystyle \int {\text{cos}}^{2}3xdx}$

${\displaystyle \int {\text{sin}}^{2}x{\text{cos}}^{2}xdx}$

${\displaystyle \int {\text{sin}}^{2}xdx}+{\displaystyle \int {\text{cos}}^{2}xdx}$

${\displaystyle \int {\text{sin}}^{2}x{\text{cos}}^2(2x)dx}$

${\displaystyle {\int }_0^{2\pi }\text{cos}x\text{sin}2xdx}$

${\displaystyle {\int }_0^{\pi }\text{sin}3x\text{sin}5xdx}$

${\displaystyle {\int }_0^{\pi }\text{cos}(99x)\text{sin}(101x)dx}$

${\displaystyle {\int }_{\text{−}\pi }^{\pi }{\text{cos}}^2(3x)dx}$

${\displaystyle {\int }_0^{2\pi }\text{sin}x\text{sin}(2x)\text{sin}(3x)dx}$

${\displaystyle {\int }_0^{4\pi }\text{cos}(x\text{/}2)\text{sin}}(x\text{/}2)dx$

${\displaystyle {\int }_{\pi \text{/}6}^{\pi \text{/}3}\frac{{\text{cos}}^{3}x}{\sqrt{\text{sin}x}}dx}$ (Round this answer to three decimal places.)

${\displaystyle {\int }_{\text{−}\pi \text{/}3}^{\pi \text{/}3}\sqrt{{\text{sec}}^{2}x-1}dx}$

${\displaystyle {\int }_0^{\pi \text{/}2}\sqrt{1-\text{cos}(2x)}dx}$

Find the area of the region bounded by the graphs of the equations $y=\text{sin}x,y={\text{sin}}^{3}x,x=0,\text{and}x=\frac{\pi }{2}.$

Find the area of the region bounded by the graphs of the equations $y={\text{cos}}^{2}x,y={\text{sin}}^{2}x,x=-\frac{\pi }{4},\text{and}x=\frac{\pi }{4}.$

A particle moves in a straight line with the velocity function $v(t)=\text{sin}(\omega t){\text{cos}}^2\left(\omega t\right).$ Find its position function $x=f(t)$ if $f(0)=0.$

Find the average value of the function $f(x)={\text{sin}}^{2}x{\text{cos}}^{3}x$ over the interval $\left[\text{−}\pi ,\pi \right].$

$\frac{dy}{dx}={\text{sin}}^{2}x.$ The curve passes through point $(0,0).$

$\frac{dy}{d\theta }={\text{sin}}^4\left(\pi \theta \right)$

Find the length of the curve $y=\text{ln}(\text{csc}x),\frac{\pi }{4}\le x\le \frac{\pi }{2}.$

Find the length of the curve $y=\text{ln}(\text{sin}x),\frac{\pi }{3}\le x\le \frac{\pi }{2}.$

Find the volume generated by revolving the curve $y=\text{cos}(3x)$ about the x -axis, $0\le x\le \frac{\pi }{36}.$

Show that $\{\text{sin}(2x),\text{cos}(3x)\}$ are orthogonal over the interval $\left[\text{−}\pi ,\pi \right].$

Evaluate ${\displaystyle {\int }_{\text{−}\pi }^{\pi }\text{sin}(mx)\text{cos}(nx)dx}.$

Integrate $y^{\prime }=\sqrt{\text{tan}x}{\text{sec}}^{4}x.$

${\displaystyle \int {\text{sin}}^{456}x\text{cos}xdx}$ or ${\displaystyle \int {\text{sin}}^{2}x{\text{cos}}^{2}xdx}$

${\displaystyle \int {\text{tan}}^{350}x{\text{sec}}^{2}xdx}$ or ${\displaystyle \int {\text{tan}}^{350}x\text{sec}xdx}$