Math 270 Name___ Lab #2
Week 2 Topics: Trigonometric Identities and powers of trigonometric functions, inverse trigonometric
functions, integration by parts, trigonometric substitution, Partial Fraction decomposition, and
integration using tables.
SHOW ALL YOUR WORK FOR FULL CREDIT, including highlighting your final answer. Note
that answers provided without supporting steps will earn approximately 1/3 credit.
1.) Integrate ∫ cos2 x sin3 x dx
∫cos^2(x)sin^3(x)dx
=∫−(sin(5x)−3(sin(3x)−sin(x))+2sin(3x)−5sin(x))/(16)dx
=−1/16∫sin(5x)dx+1/16∫sin(3x)dx+1/8∫sin(x)dx
=cos(5x)/80−cos(3x)/48−cos(x)/8+C
=[3cos^5(x)−5cos^3(x)]/[15]+C
=[cos^3(x)(3cos^2(x)-5)]/[15]+C
2.) Integrate
∫sin^5(x)dx
=∫(1−cos^2(x))^2 sin(x)dx
=∫−(1−u^2)^2du
=−∫(u^2−1)^2du
=∫(u^4−2u^2+1)du
=∫u^4du−2∫u^2du+∫1du
=u^5/5−2u^3/3+u
=−[cos^5(x)/5]+[2cos^3(x)/3]−cos(x)
=−[cos^5(x)/5]+[2cos^3(x)/3]−cos(x)+C
3.) Integrate
∫x cos^2(2x^2)dx
=1/4∫cos^2(u)du
=[cos(u)sin(u)]/[2]+1/2∫1du
=[cos(u)sin(u)]/[2]+u/2
=[cos(u)sin(u)]/[8]+u/8
=[cos(2x^2)sin(2x^2)]/[8]+[x^2]/[4]+C
=[sin(4x^2)+4x^2]/[16]+C
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, Math 270 Name___ Lab #2
4.) Integrate
∫sec^2(x) tan^5(x) dx
=∫ u^5 du
=u^6/6
=tan^6(x)/6
=[tan^6(x)]/[6]+C
5.) Integrate
∫tan^2(x)dx
=∫(sec^2(x)−1)dx
=∫sec^2(x)dx−∫1dx
=tan(x)−x
=tan(x)−x+C
6.) Using the list:
(a) inverse sine; (b) inverse tangent; (c) logarithmic; (d) general power; or (e) none of these,
identify the form of each of the following integrals. Do not integrate.
= ___C____ = ____B___ = ___C____
= _A__ = __C_ = ___A____
= __A___ = __D__ = ___D____
Page 2
Week 2 Topics: Trigonometric Identities and powers of trigonometric functions, inverse trigonometric
functions, integration by parts, trigonometric substitution, Partial Fraction decomposition, and
integration using tables.
SHOW ALL YOUR WORK FOR FULL CREDIT, including highlighting your final answer. Note
that answers provided without supporting steps will earn approximately 1/3 credit.
1.) Integrate ∫ cos2 x sin3 x dx
∫cos^2(x)sin^3(x)dx
=∫−(sin(5x)−3(sin(3x)−sin(x))+2sin(3x)−5sin(x))/(16)dx
=−1/16∫sin(5x)dx+1/16∫sin(3x)dx+1/8∫sin(x)dx
=cos(5x)/80−cos(3x)/48−cos(x)/8+C
=[3cos^5(x)−5cos^3(x)]/[15]+C
=[cos^3(x)(3cos^2(x)-5)]/[15]+C
2.) Integrate
∫sin^5(x)dx
=∫(1−cos^2(x))^2 sin(x)dx
=∫−(1−u^2)^2du
=−∫(u^2−1)^2du
=∫(u^4−2u^2+1)du
=∫u^4du−2∫u^2du+∫1du
=u^5/5−2u^3/3+u
=−[cos^5(x)/5]+[2cos^3(x)/3]−cos(x)
=−[cos^5(x)/5]+[2cos^3(x)/3]−cos(x)+C
3.) Integrate
∫x cos^2(2x^2)dx
=1/4∫cos^2(u)du
=[cos(u)sin(u)]/[2]+1/2∫1du
=[cos(u)sin(u)]/[2]+u/2
=[cos(u)sin(u)]/[8]+u/8
=[cos(2x^2)sin(2x^2)]/[8]+[x^2]/[4]+C
=[sin(4x^2)+4x^2]/[16]+C
Page 1
, Math 270 Name___ Lab #2
4.) Integrate
∫sec^2(x) tan^5(x) dx
=∫ u^5 du
=u^6/6
=tan^6(x)/6
=[tan^6(x)]/[6]+C
5.) Integrate
∫tan^2(x)dx
=∫(sec^2(x)−1)dx
=∫sec^2(x)dx−∫1dx
=tan(x)−x
=tan(x)−x+C
6.) Using the list:
(a) inverse sine; (b) inverse tangent; (c) logarithmic; (d) general power; or (e) none of these,
identify the form of each of the following integrals. Do not integrate.
= ___C____ = ____B___ = ___C____
= _A__ = __C_ = ___A____
= __A___ = __D__ = ___D____
Page 2
