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Solutions to various integration problems and substitutions covered in a math 1b course. Topics include using integration by parts, evaluating integrals of trigonometric functions, and making substitutions. Students will find worked-out examples and practice exercises.
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Wednesday January 23, 2008 Math 1B
(ln x)n−^1 dx.
b. Evaulate
(ln x)^3 dx.
(a)
∫ (^) π 0 sin
(^3) x cos (^8) x dx
(b)
∫ (^) π 0 sin
(^4) x dx
(c)
sin^2 x cos^2 x dx (d)
tan^2 x sin^3 x dx
(a)
∫ (^) cos x+ cos x− 1 dx (b)
∫ (^) sin x+cos x sin 2x dx
(a)
9 − e^2 tdx (b)
∫ (^) dx √x (^2) − 4 x− 5
(c)
∫ (^) dx x+x^3.
a^2 − x^2.
(a) For which values of x is f (x) defined? Sketch the domain of f on a number line. (b) Draw a right triangle and decide which edges best represent x and f (x). Label all three edges with an appropriate value. Express sin θ, tan θ and sec θ (where θ is an acute angle of your triangle) in terms of the values written on the edges. (c) Write x as a function of θ, x = j(θ). What is the domain and range of the function j? (d) Does the function f (j(θ)) have the same domain and range as the function f (x)? (e) Now integrate
∫ (^) dx f (x) using the substitution^ x^ =^ j(θ).
