Math 141 Exam 1 Questions with100% Correct
Answers
integration by parts
∫udv = uv - ∫v*du
∫sinmxcosnxdx when m is odd
Save one sine factor
Use substitution: u = cosx, du = -sinx*dx
Use the identity: sin2x = 1 - cos2x = 1 - u2
∫sinmxcosnxdx when n is odd
Save one cosine factor
Use substitution: u = sinx, du = cosx*dx
Use the identity: cos2x = 1 - sin2x = 1 - u2
∫sinmxcosnxdx when m and n are both even
Use the half-angle formulas: sin2x = ½ (1-cos2x) and cos2x = ½ (1+cos2x)
Use ∫cos(kx)dx = 1/k sin(kx) +C
∫tanmxsecnxdx when m is odd
Save one secx*tanx factor
Use substitution: u = secx, du = secxtanx*dx
Use identity: tan2x = sec2x - 1 = u2 - 1
∫tanmxsecnxdx when n is even
, Save one sec2x factor
Use substitution: u = tanx and du = sec2xdx
Use identity: sec2x = 1 + tan2x = 1 + u2
∫tanmxsecnxdx when m is even and n is odd
Use the identity: tan2x = sec2x - 1
trigonometric substitution for √(a^2-x^2)
x = asinθ
trigonometric substitution for √(a^2+x^2)
x = atanθ
trigonometric substitution for √(x^2-a^2)
x = asecθ
integration by partial fractions when the denominator Q(x) is a product of distinct linear
factors
Aln|a1x+b1| + Bln|a2x+b2|...
integration by partial fractions when the denominator Q(x) is a product of factors, some of
which are repeated
Aln|a1x+b1| - B/(a1x+b1)...
integration by partial fractions when Q(x) contains irreducible quadratic factors, none of
which are repeated
Answers
integration by parts
∫udv = uv - ∫v*du
∫sinmxcosnxdx when m is odd
Save one sine factor
Use substitution: u = cosx, du = -sinx*dx
Use the identity: sin2x = 1 - cos2x = 1 - u2
∫sinmxcosnxdx when n is odd
Save one cosine factor
Use substitution: u = sinx, du = cosx*dx
Use the identity: cos2x = 1 - sin2x = 1 - u2
∫sinmxcosnxdx when m and n are both even
Use the half-angle formulas: sin2x = ½ (1-cos2x) and cos2x = ½ (1+cos2x)
Use ∫cos(kx)dx = 1/k sin(kx) +C
∫tanmxsecnxdx when m is odd
Save one secx*tanx factor
Use substitution: u = secx, du = secxtanx*dx
Use identity: tan2x = sec2x - 1 = u2 - 1
∫tanmxsecnxdx when n is even
, Save one sec2x factor
Use substitution: u = tanx and du = sec2xdx
Use identity: sec2x = 1 + tan2x = 1 + u2
∫tanmxsecnxdx when m is even and n is odd
Use the identity: tan2x = sec2x - 1
trigonometric substitution for √(a^2-x^2)
x = asinθ
trigonometric substitution for √(a^2+x^2)
x = atanθ
trigonometric substitution for √(x^2-a^2)
x = asecθ
integration by partial fractions when the denominator Q(x) is a product of distinct linear
factors
Aln|a1x+b1| + Bln|a2x+b2|...
integration by partial fractions when the denominator Q(x) is a product of factors, some of
which are repeated
Aln|a1x+b1| - B/(a1x+b1)...
integration by partial fractions when Q(x) contains irreducible quadratic factors, none of
which are repeated
