삼각함수의 적분

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

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

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

For the following exercises, solve the differential equations.

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

For the following exercises, solve the differential equations.

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

For the following exercises, solve the differential equations.

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

For the following exercises, solve the differential equations.

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

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

For the following exercises, use this information: The inner product of two functions f and g over $[a,b]$ is defined by $f(x)\cdot g(x)=\langle f,g\rangle ={\int }_{a}^{b}f\cdot gdx.$ Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0.$

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

For the following exercises, use this information: The inner product of two functions f and g over $[a,b]$ is defined by $f(x)\cdot g(x)=\langle f,g\rangle ={\int }_{a}^{b}f\cdot gdx.$ Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0.$

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

For the following exercises, use this information: The inner product of two functions f and g over $[a,b]$ is defined by $f(x)\cdot g(x)=\langle f,g\rangle ={\int }_{a}^{b}f\cdot gdx.$ Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0.$

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

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.

$\int {\sin }^{456}x\cos x\,dx$ or $\int {\sin }^{2}x{\cos }^{2}x\,dx$

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.

$\int {\tan }^{350}x{\sec }^{2}x\,dx$ or $\int {\tan }^{350}x\sec x\,dx$

T/F: $\displaystyle \int \sin^2x\cos^2x \ dx$ cannot be evaluated using the techniques described in this section since both powers of $\sin x$ and $\cos x$ are even.

T/F: $\displaystyle \int \sin^3x\cos^3x \ dx$ cannot be evaluated using the techniques described in this section since both powers of $\sin x$ and $\cos x$ are odd.

T/F: This section addresses how to evaluate indefinite integrals such as $\displaystyle \int \sin^5x\tan^3x\ dx.$

T/F: Sometimes computer programs evaluate integrals involving trigonometric functions differently than one would using the techniques of this section. When this is the case, the techniques of this section have failed and one should only trust the answer given by the computer.

In Exercises 5–28, evaluate the indefinite integral.

$\displaystyle \int \sin x\cos^4x\ dx$

In Exercises 5–28, evaluate the indefinite integral.

$\displaystyle \int \sin^3 x\cos x\ dx$

In Exercises 5–28, evaluate the indefinite integral.

$\displaystyle \int \sin^3 x\cos^2 x\ dx$