Section 7.2: Trigonometric Integration

Review

We have three antidifferentiation techniques:

Recognize it immediately as the derivative of some function, for example
	∫(e^x)d x=(e^x)+C

Try substituting for some internal part. Don’t forget the differential. For example, to evaluate ∫(x^2)cos((x^3))d x use the substitution u=(x^3) and d u=3(x^2)d x to replace the above integral with the simpler one
	∫(1/3)cos(u)d u=(1/3)sin(u)+C=(1/3)sin((x^3))+C

Integration by parts:
	∫u d v=u v-∫v d u

Questions

Choose an appropriate technique to evaluate the following integrals.

Find the area of the region bounded above by y=x (e^(-x)),  below by the x-axis, and the line x=5.

Plot[x E^-x,{x,-1,5}]

Find the volume of the surface of revolution generated by rotating about the x-axis the region bounded above by y=x sec((x^3)),  below by the x-axis, and the line x=1.

Plot[x Sec[(x^3)],{x,-Pi/2,Pi/2}]
Integrate[Pi x^2 Sec[(x^3)]^2,{x,0,1}]
N[%]

Trigonometric identities

Pythagorean identities

(cos^2)(θ)+(sin^2)(θ)=1

Questions

Divide this equation by (cos^2)(θ) to get another Pythagorean identity.

Divide this equation by (sin^2)(θ) to get another Pythagorean identity.

Double angle formulas

The sine and cosine sum formulas are very important:
	sin(α+β)=sin(α)cos(β)+sin(β)cos(α)
cos(α+β)=cos(α)cos(β)-sin(α)sin(β).

Questions

What is sin(2θ)=sin(θ+θ)?

What is cos(2θ)=cos(θ+θ)?

Rewrite this using a Pythagorean trigonometric identity so that cos(2θ) is in terms of only cos(θ).

Rewrite this using a Pythagorean trigonometric identity so that cos(2θ) is in terms of only sin(θ).

Trigonometric integrals

Questions

What are the following derivatives?

(d/(d x))cos(x)

(d/(d x))sin(x)

(d/(d x))tan(x)

(d/(d x))sec(x)

D[Sec[x],x]

(d/(d x))cot(x)

D[Cot[x],x]

(d/(d x))csc(x)

D[Csc[x],x]

Questions

Evaluate ∫(cos^4)(x)sin(x)d x using the substitution u=cos(x).

Integrate[Cos[x]^4 Sin[x],x]

Evaluate ∫(sin^6)(x)(cos^3)(x)d x by writing this as ∫(sin^6)(x)(cos^2)(x)cos(x)d x=∫(sin^6)(x)(1-(sin^2)(x))cos(x)d x.

Integrate[Sin[x]^6 Cos[x]^3 ,x]
Integrate[(Sin[x]^6(1-Sin[x]^2))Cos[x],x]

Find the area under the curve y=(sin^3)(x)(cos^3)(x) for 0≤x≤π/2.

Plot[Sin[x]^3 Cos[x]^3,{x,0,Pi/2}]

Evaluate ∫(cot^4)(x)(csc^2)(x)d x.

Integrate[Cot[x]^4 Csc[x]^2,x]

Evaluate ∫(tan^2)(x)(sec^4)(x)d x.

Integrate[Tan[x]^2 Sec[x]^4,x]

Questions

Since cos(2θ)=2(cos^2)(θ)-1, solve for (cos^2)(θ) to get (cos^2)(θ)=((1+cos(2θ))/2). Use this to evaluate
	∫(cos^2)(θ)d θ=∫((1+cos(2θ))/2)d θ.

Find the volume of the solid of revolution obtained by rotating about the x-axis the region above the x-axis and one arch of the sine curve y=sin(x).

Plot[Sin[x],{x,0,Pi}]