MAC 2312 Lecture Notes - Lecture 5: Deuterium

Indefinite Integrals, with substitution 6. Evaluate the following indefinite integrals. These require substitution. Show the substitution. For example, for f(x)=f8x cosh (4x2)dx f'( The substitution is u=4x2 . The derivative is a 8xdr . Substitute the u and the du into the integral. In this case, replace the 8xdx with du. Do not use an intermediate step with both old and new variables in the integral. x)=8xcosh(4x2), then f(u)-I cosh u du=sinh u+ C Reverse the substitution: ,f(x)=sinh 4x4 C 0.5 | | sin-111-cos-ldt u=1-cos- 2

1. Find the integrals using tabular method a) J cos 2z dzbd 2. Find the indefinite integrals using integration by parts a) Jedr sin(3r)dr 3. Find the following derivatives a) f(z) = tanh(3rs_ 1) b) g(z) = zsinh(z) _ cosh(z) c) h(z)-y?coth"'- d)y = sech(cos 2z) f) f(t) sech-1(2c) 4. Find the following integrals c/sech3ltanhzdz d)/å ­Hint: using completing square 5. a) Evaluate the integral fo9rd using Special Trig Substitution formula. 6. Find the following trigonometric integrals a) fcoss () sin (x) drb)f'sin5 cos 3z dr c) sees(t) tan'()dt 7. Use trigonometric substitution to set up the integrals ar- dr. Do Not Integrate. b) Use Trigonometric guidelines to find new integral from part (a). Do Not forget to change back to the original variable. 8. Find the following integrals using partial fractions.

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