9B-无代写
时间:2023-05-18
Math 9B Worksheet 7 Spring 2023
1. (a) Now you know even more techniques for integration! Explain the different clues for
when to use integration by parts, trig substitution, substitution, or partial fractions.
(b) Evaluate
∫ 2
1
ln(x)
(x− 3)2 dx.
(c) True or False: the integral
∫
x3√
x2 − 7 dx can be solved by regular u-substitution as
well as with trig substitution. Justify your answer with calculations.
2. (a) Why is there not a separate rule (in the partial fraction decomposition key idea) for
what to do if you have a cubic function in the denominator of a rational function?
(b) Definite integrals which use trig substitution can be tricky. Evaluate the following
integral
∫ −2/5
−4/5
√
25x2 − 4
x
dx. How is this different than the integral on [2/5, 4/5]?
Hint: it has to do with absolute value.
(c) Rewrite the following rational function using partial fraction decomposition. Please
don’t solve for the constants.
3x2 − 5x+ 7
(x2 + x− 2)2(x− 3)4(x2 + 5)(x2 − 2x+ 17)2
3. (optional extra practice) You don’t need to turn these in.
∫ 4
3
x3 − 2x2 − 4
x3 − 2x2 dx
∫
x3 + x2 + 2x+ 1
(x2 + 1)(x2 + 2)
dx
∫ 1
0
x
√
x2 + 4 dx
∫
ln(x2 + x+ 1)
x2
dx
∫ 3
2
1
x2 − 1 dx
∫
x√
x2 + x+ 1
dx