MATH 142 Final Exam: Integration Techniques and Trigonometric Substitutions, Exams of Mathematics

Download MATH 142 Final Exam: Integration Techniques and Trigonometric Substitutions and more Exams Mathematics in PDF only on Docsity! MATH 142 – Final Exam √ a2 − x2 → x = a sin t − π 2 ≤ t ≤ π 2 → a2−x2 = a2−a2 sin2 t = a2 cos2 t √ a2 + x2 → x = a tan t − π 2 < t < π 2 → a2+x2 = a2+a2 tan2 t = a2 sec2 t √ x2 − a2 → x = a sec t { 0 ≤ t < π/2 if x ≥ a π/2 < t ≤ π if x ≤ a → x 2− a2 = a2 sec2 t− a2 = a2 tan2 t ∫ tanm x secn x dx ⇒    n even → u = tanx → sec2 x = tan2 x+ 1 m odd → u = sec x → tan2 x = sec2 x− 1 m even and n odd → tan2 x = sec2 x− 1 ∫ sinm x cosn x dx ⇒    n odd → u = sin x → cos2 x = 1− sin2 x m odd → u = cos x → sin2 x = 1− cos2 x m even and n even → { sin2 x = 1 2 (1− cos 2x) cos2 x = 1 2 (1 + cos 2x) sinα cos β = 1 2 [sin(α− β) + sin(α + β)] sinα sin β = 1 2 [cos(α− β)− cos(α + β)] cosα cos β = 1 2 [cos(α− β) + cos(α + β)] sin(α + β) = sinα cos β + cosα sin β sin(α− β) = sinα cos β − cosα sin β cos(α + β) = cosα cos β + sinα sin β cos(α− β) = cosα cos β − sinα sin β sin(2θ) = 2 sin θ cos θ sin2 θ + cos2 θ = 1 ∫ sinn x dx = − 1 n sinn−1 x cosx+ n− 1 n ∫ sinn−2 x dx ∫ cosn x dx = 1 n cosn−1 x sinx+ n− 1 n ∫ cosn−2 x dx