9.3.3 More Integrating Tirgonometric Functions by Substitution Flashcards
1
Q
More Integrating Trigonometric Functions by Substitution
A
- You can apply integration by substitution to integrands involving trigonometric functions that are not composite functions.
- When working with integrands that include trigonometric expressions, it is sometimes necessary to rewrite those expressions using trig identities.
2
Q
note
A
- Instead of a composite function, this integral involves the product of two trigonometric functions.
- You could let u be sinx, in which case du would be cosx, or you could let u be cosx, making du be –sinx. You might want to choose u = sinx to avoid the negative sign.
- Once you have determined the expression for u, the integrand should be simple to evaluate. Remember to replace u with its expression in terms of x.
- You can check your work by integrating with the help of the chain rule.
- You may often find it useful to express trigonometric
integrands in terms of the sine and cosine functions. - Notice that you must choose u = cosx, since it is in the denominator. That way the du-expression can replace the numerator and dx.
- Factor out the –1 from the integrand.
- The integral of du/u is ln|u| + C.
- Make sure to express your result in terms of x.
- Check that your answer is correct by integrating.
3
Q
Evaluate the integral.
∫secxtanx√1+secxdx
A
2/3(1+secx)^(3/2)+C
4
Q
Integrate.∫cotxdx
A
ln | sin x | + C
5
Q
Integrate.∫csc^2t / tan^2t dt
A
−cot^3t/3+C
6
Q
Evaluate.∫8sin^33xcos3xdx
A
2/3sin^4 3x+C
7
Q
Integrate.∫cos^4(x^2)sin(x^2)2xdx
A
−cos^5x^2/5+C
8
Q
Integrate.∫tanxdx
A
ln | sec x | + C
9
Q
Evaluate.∫tan^2θ/cos^2θdθ
A
tan^3θ/3+C
10
Q
Integrate.∫sec(sinx)tan(sinx)cosxdx
A
sec (sin x) + C
11
Q
Evaluate.∫sinxcosxdx
A
sin^2x/2+C
12
Q
Evaluate the integral.∫cosxcos(sinx) dx
A
sin (sin x) + C
