# 4 trigonometric substitutions 75 integrating rational

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7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.8 Improper Integrals 7.7 Numerical Integration; Simpson’s Rule Example 12; An Alternate Approach sin x cos x - sin x dx = 1 2 sin x + 1 2 cos x + 1 2 sin x - 1 2 cos x cos x - sin x dx = 1 2 sin x + cos x cos x - sin x dx + 1 2 sin x - cos x cos x - sin x dx = - 1 2 1 u du - 1 2 dx , with u = cos x - sin x = - 1 2 ln | u | - 1 2 x + C = - 1 2 ln | cos x - sin x | - 1 2 x + C This gives a different answer; but both are correct. Chapter 7 Lecture Notes MAT187H1F Lec0101 Burbulla Chapter 7: Principles of Integral Evaluation 7.1 An Overview of Integration Methods 7.2 Integration by Parts 7.3 Integrating Trigonometric Functions 7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.8 Improper Integrals 7.7 Numerical Integration; Simpson’s Rule Chapter 7 Lecture Notes MAT187H1F Lec0101 Burbulla
Chapter 7: Principles of Integral Evaluation 7.1 An Overview of Integration Methods 7.2 Integration by Parts 7.3 Integrating Trigonometric Functions 7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.8 Improper Integrals 7.7 Numerical Integration; Simpson’s Rule Trigonometric Substitutions; aka Inverse Trig Substitutions Integrand Contains Trig Substitution Inverse Trig Substitution a 2 - x 2 try x = a sin θ or θ = sin - 1 x a a 2 + x 2 try x = a tan θ or θ = tan - 1 x a x 2 - a 2 try x = a sec θ or θ = sec - 1 x a Chapter 7 Lecture Notes MAT187H1F Lec0101 Burbulla Chapter 7: Principles of Integral Evaluation 7.1 An Overview of Integration Methods 7.2 Integration by Parts 7.3 Integrating Trigonometric Functions 7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.8 Improper Integrals 7.7 Numerical Integration; Simpson’s Rule Example 1; x = tan θ ; dx = sec 2 θ d θ 1 (1 + x 2 ) 3 / 2 dx = 1 (1 + tan 2 θ ) 3 / 2 sec 2 θ d θ = 1 (sec 2 θ ) 3 / 2 sec 2 θ d θ = sec 2 θ sec 3 θ d θ = 1 sec θ d θ = cos θ d θ = sin θ + C Chapter 7 Lecture Notes MAT187H1F Lec0101 Burbulla
Chapter 7: Principles of Integral Evaluation 7.1 An Overview of Integration Methods 7.2 Integration by Parts 7.3 Integrating Trigonometric Functions 7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.8 Improper Integrals 7.7 Numerical Integration; Simpson’s Rule Example 1, Continued So far we have 1 (1 + x 2 ) 3 / 2 dx = sin θ + C . At this point two comments are in order: 1. In simplifying (sec 2 θ ) 3 / 2 = sec 3 θ, we assumed sec θ > 0 . Unless we have information otherwise, this will be our usual approach in simplifying trig integrals after making a trig substitution.