Solution: Integrals of the form sin mx cos nx dx Use the

State the half-angle identities used to integrate \(\sin ^{2} x\) and \(\cos ^{2} x\)

State the three Pythagorean identities.

Describe the method used to integrate \(\sin ^{3} x\).

Describe the method used to integrate \(\sin ^{m} x \cos ^{n} x\) for m even and n odd.

What is a reduction formula?

How would you evaluate \(\int \cos ^{2} x \sin ^{3} x d x\)?

How would you evaluate \(\int \tan ^{10} x \sec ^{2} x d x\)?

How would you evaluate / secl x tan x dx?

Integrals of sin x or cos x Evaluate the following integrals \(\int \sin ^{2} x d x\)

Integrals of sin x or cos x Evaluate the following integrals \(\int \cos ^{4} 2 x d x\)

Integrals of sin x or cos x Evaluate the following integrals \(\int \sin ^{5} x d x)

Integrals of sin x or cos x Evaluate the following integrals \(\int \cos ^{3} 20 x d x\)

Integrals of sin x and cos x Evaluate the following integrals \(\int \sin ^{2} x \cos ^{2} x d x\)

Integrals of sin x and cos x Evaluate the following integrals \(\int \sin ^{3} x \cos ^{5} x d x\)

Integrals of sin x and cos x Evaluate the following integrals \(\int \sin ^{5} x \cos ^{-2} x d x\)

Integrals of sin x and cos x Evaluate the following integrals \(\int \sin ^{-3 / 2} x \cos ^{3} x d x)

Integrals of sin x and cos x Evaluate the following integrals \(\int \sin ^{2} x \cos ^{4} x d x\)

Integrals of sin x and cos x Evaluate the following integrals \(\int \sin ^{3} x \cos ^{3 / 2} x d x\)

Integrals of tan x or cot x Evaluate the following integrals \(\int \tan ^{2} x d x\)

Integrals of tan x or cot x Evaluate the following integrals \(\int 6 \sec ^{4} x d x)

Integrals of tan x or cot x Evaluate the following integrals \(\int \tan ^{3} 4 x d x\)

Integrals of tan x or cot x Evaluate the following integrals \(\int \sec ^{5} \theta d \theta\)

Integrals of tan x or cot x Evaluate the following integrals \(\int 20 \tan ^{6} x d x\)

Integrals of tan x or cot x Evaluate the following integrals \(\int \cot ^{5} 3 x d x\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int \sec ^{2} x \tan ^{1 / 2} x d x\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int \sec ^{-2} x \tan ^{3} x d x\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int \frac{\csc ^{4} x}{\cot ^{2} x} d x\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int \csc ^{10} x \cot ^{3} x d x\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int_{0}^{\pi / 4} \sec ^{4} \theta d \theta\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int \tan ^{5} \theta \sec ^{4} \theta d \theta\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int_{\pi / 6}^{\pi / 3} \cot ^{3} \theta d \theta\)

Integrals of tan x and sec x Evaluate the following integrals. \(\int_{0}^{\pi / 4} \tan ^{3} \theta \sec ^{2} \theta d \theta\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample, a. If m is a positive integer, then \(\int_{0}^{\pi} \cos ^{2 m+1} x d x=0\). b. If m is a positive integer, then \(\int_{0}^{\pi} \sin ^{m} x d x=0 \text {. }\)

Integrals of cot x, sec x, and csc x Use a change of variables to prove that \(\int \cot x d x=\ln |\sin x|+C\)

Integrals of cot x, sec x, and csc x Prove that \(\int \sec x d x=\ln |\sec x+\tan x|+C\). (Hint: Multiply numerator and denominator of the integrand by secx + tan x; then make a change of variables with u = sec x + tan x.)

Integrals of cot x, sec x, and csc x Prove that \(\int \csc x d x=-\ln |\csc x+\cot x|+C\). (Hint: Use a method analogous to that used in Exercise 35.)

Integrals of cot x, sec x, and csc x Use the results of Theorem 7.1 to find the indefinite integral of tan ax and sec ax, where a is a nonzero real number.

Comparing areas The region \(R_{1}\), is bounded by the graph of y = tan x and the x-axis on the interval (0,\(\pi\)/3). The region \(R_{2}\), is bounded by the graph of y = sec x and the x-axis on the interval (0,\(\pi\)/6). Which region has the greater area?

Region between curves Find the area of the region bounded by the graphs of y = tan x and y = sec x on the interval (0,\(\pi\)/4).

Additional integrals Evaluate the following integrals. \(\int_{0}^{\sqrt{\pi / 2}} x \sin ^{3}\left(x^{2}\right) d x\)

Additional integrals Evaluate the following integrals. \(\int \frac{\sec ^{4}(\ln \theta)}{\theta} d \theta\)

Additional integrals Evaluate the following integrals. \(\int_{\pi / 6}^{\pi / 2} \frac{d y}{\sin y}\)

Additional integrals Evaluate the following integrals. \(\int_{-\pi / 3}^{\pi / 3} \sqrt{\sec ^{2} \theta-1} d \theta\)

Additional integrals Evaluate the following integrals. \(\int_{-\pi / 4}^{\pi / 4} \tan ^{3} x \sec ^{2} x \ d x\)

Additional integrals Evaluate the following integrals. \(\int_{0}^{\pi}(1-\cos 2 x)^{3 / 2} \ d x\)

Square roots Evaluate the following integrals \(\int_{-\pi / 4}^{\pi / 4} \sqrt{1+\cos 4 x} \ d x\)

Square roots Evaluate the following integrals \(\int_{0}^{\pi / 2} \sqrt{1-\cos 2 x} \ d x\)

Square roots Evaluate the following integrals \(\int_{0}^{\pi / 8} \sqrt{1-\cos 8 x} \ d x\)

Square roots Evaluate the following integrals \(\int_{0}^{\pi / 4}(1+\cos 4 x)^{3 / 2} d x\)

Sine football Find the volume of the solid generated when the region bounded by the graph of y = sin x and the x-axis on the interval [0,\(\pi\) is revolved about the x-axis.

Arc length Find the length of the curve y = ln(cos x) fir \(0 \leq x \leq \pi/4\)

A sine reduction formula Use integration by parts to obtain the following reduction formula for positive integers n: \(\int \sin ^{n} x \ d x=-\sin ^{n-1} x \cos x+(n-1) \int \sin ^{n-2} x \cos ^{2} x \ d x\) Then use an identity to obtain the reduction formula \(\int \sin ^{n} x \ d x=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} \int \sin ^{n-2} x \ d x .\) Use this reduction formula to evaluate \(\int \sin ^{6} x \ d x\).

A tangent reduction formula Prove that for positive integers \(n \neq 1\) \(\int \tan ^{n} x \ d x=\frac{\tan ^{n-1} x}{n-1}-\int \tan ^{n-2} x \ d x\) Use the formula to evaluate \(\int_{0}^{\pi / 4} \tan ^{3} x \ d x\)

A secant reduction formula Prove that for positive integers \(n \neq 1\), \(\int_{ }^{ }\sec^nx\ dx=\frac{\sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int_{ }^{ }\sec^{n-2}x\ dx\) (Hint: Integrate by parts with \(u=\sec ^{n-2} x\) and \(dv=\sec^2x\ dx\)

Integrals of the form \(\int \sin m x \cos n x \ d x\) Use the following three identities to evaluate the given integrals. \(\sin m x \sin n x =\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\) \(\sin m x \cos n x =\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\) \(\cos m x \cos n x =\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\) \(\int \sin 3 x \cos 7 x \ d x\)

Integrals of the form \(\int \sin m x \cos n x \ d x\) Use the following three identities to evaluate the given integrals. \(\sin m x \sin n x =\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\) \(\sin m x \cos n x =\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\) \(\cos m x \cos n x =\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\) \(\int \sin 5 x \sin 7 x \ d x\)

Integrals of the form \(\int \sin m x \cos n x \ d x\) Use the following three identities to evaluate the given integrals. \(\sin m x \sin n x =\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\) \(\sin m x \cos n x =\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\) \(\cos m x \cos n x =\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\) \(\int \sin 3 x \sin 2 x \ d x\)

Integrals of the form \(\int \sin m x \cos n x d x\) Use the following three identities to evaluate the given integrals. \(\sin m x \sin n x =\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\) \(\sin m x \cos n x =\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\) \(\cos m x \cos n x =\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\) \(\int \cos x \cos 2 x d x\)

Integrals of the form \(\int \sin m x \cos n x d x\) Use the following three identities to evaluate the given integrals. \(\sin m x \sin n x =\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\) \(\sin m x \cos n x =\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\) \(\cos m x \cos n x =\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\) Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with \(m \neq n\). (a) \(\int_{0}^{\pi} \sin m x \sin n x d x=0\) (b) \(\int_{0}^{\pi} \cos m x \cos n x d x=0\) (c) \(\int_{0}^{\pi} \sin m x \cos n x d x=0\)

Mercator map projection The Mercator map projection was proposed by the Flemish geographer Gerardus Mercator (1512-1594). The stretching of the Mercator map as a function of the latitude \(\theta\) is given by the function \(G(\theta)=\int_{0}^{\theta} \sec x \ d x\) Graph G for \(0 \leq \theta<\pi / 2\). (See the Guided Projects for a derivation of this integral.)

Exploring powers of sine and cosine (a) Graph the functions \(f_{1}(x)=\sin ^{2} x\) and \(f_{2}(x)=\sin ^{2} 2 x\) on the interval \([0, \pi]\). Find the area under these curves on \([0, \pi]\). (b) Graph a few more of the functions \(f_{n}(x)=\sin ^{2} n x\) on the interval \([0, \pi]\), where n is a positive integer. Find the area under these curves on \([0, \pi]\). Comment on your observations. (c) Prove that \(\int_{0}^{\pi} \sin ^{2}(n x) \ d x\) has the same value for all positive integers n. (d) Does the conclusion of part (c) hold if sine is replaced by cosine? (e) Repeat parts (a), (b), and (c) with \(\sin ^{2} x\) replaced by \(\sin ^{4} x\). Comment on your observations. (f) Challenge problem: Show that for \(m=1,2,3, \ldots\), \(\int_{0}^{\pi} \sin ^{2 m} x \ d x=\int_{0}^{\pi} \cos ^{2 m} x \ d x=\pi \cdot \frac{1 \cdot 3 \cdot 5 \cdots(2 m-1)}{2 \cdot 4 \cdot 6 \cdots 2 m}\)