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course Mth 174
Section 7.21)7.2.6 Find the integral Int(xe^(-2x) dx).
Integral = (-1/4)e^(-2x)*(2x+1) + c
2) 7.2.10 Find the integral Int(q^4 ln(4q) dq).
Integral = (1/25)*q^5*(5log(4q) -1 ) + c
3) 7.2.12 Find the integral Int(sin^2(x) dx).
Integral = (-1/2) (sin(x) * cos(x)) + x/2 + c
4) 7.2.16 Find the integral Int((t+2)sqrt(2+3t) dt).
Integral = (8/27)* (3t+2)^(3/2) + (2/45)(2+3t)^(5/2) + c
5) 7.2.18 Find the integral Int(z^2/e^z dz).
Integral = -e^(-z)* (z^2+2z + 2) + c
6) 7.2.20 Find the integral Int(3y/sqrt(4+y) dy).
Integral = 2(y - 8)*sqrt(y+4) + c
7) 7.2.24 Find the integral Int(arctan 4z dz)
Integral = z*arctan(4z) - (1/8)log(16z^2 + 1) + c
8) 7.2.27 Find the integral Int(x^5 cos(x^3) dx).
Integral = (1/3)(x^3*sin(x)^3)+ cos(x)^3) + c
9) 7.2.30 Evaluate the definite integral Int(ln t dt, 2, 4).
Integral = x ln(x) - x + c
(4ln(4)-4) - (2ln(2)- 2) = ~1.55 - (-0.614) = about 2.16
10) 7.2.36 Evaluate the definite integral Int(arccos z dz, -1, 1).
Integral = z* arcos(z) - sqrt( 1- z^2) + c
-pi - 0 = -pi
11) 7.2.40 In number 3) you found Int(sin^2(x) dx) using integration by parts most likely, if not then do so. Redo this integral using the identity sin^2(x) = (1 - cos(2x))/2. Explain any differences in the form of the solution found by the two methods.
Int(sin^2(x) dx), so Int(1-cos(2x))/2
Integral = 1/2 ( x - sin x cos x) + c, while the solution from number 3 was:
(-1/2) (sin(x) * cos(x)) + x/2 + c
The solution from number 3 has a (-1/2) because of the way it is worked out in parts. Since the first integral above was worked out using identities instead of in parts and using u substitution, there is a slight variation in the answers.
12) 7.2.42 Find the integral Int(e^x(sin x) dx) (Do integration by parts twice and go from there).
Integral = e^x (sin x - cos x)/2
13) 7.2.50 Given f(0) = 6, f(1) = 5, f '(1)=2. Find Int(x * f '(x), x, 0, 1).
f(x) = -x + 6
Ingetral = (1/2) x^2 * (-1/2)x^2
0 - (-1/4) = 1/4
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