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MATH 142 – Final Exam
a^2 − x^2 → x = a sin t − π 2
≤ t ≤ π 2
→ a^2 − x^2 = a^2 − a^2 sin^2 t = a^2 cos^2 t
√ a^2 + x^2 → x = a tan t − π 2 < t < π 2 → a^2 +x^2 = a^2 +a^2 tan^2 t = a^2 sec^2 t
√ x^2 − a^2 → x = a sec t
0 ≤ t < π/ 2 if x ≥ a π/ 2 < t ≤ π if x ≤ a →^ x
(^2) − a (^2) = a (^2) sec (^2) t − a (^2) = a (^2) tan (^2) t
tanm^ x secn^ x dx ⇒
n even → u = tan x → sec^2 x = tan^2 x + 1 m odd → u = sec x → tan^2 x = sec^2 x − 1 m even and n odd → tan^2 x = sec^2 x − 1
sinm^ x cosn^ x dx ⇒
n odd → u = sin x → cos^2 x = 1 − sin^2 x m odd → u = cos x → sin^2 x = 1 − cos^2 x m even and n even →
sin^2 x = 12 (1 − cos 2x) cos^2 x = 12 (1 + cos 2x)
sin α cos β =
[sin(α − β) + sin(α + β)]
sin α sin β =
[cos(α − β) − cos(α + β)]
cos α cos β =
[cos(α − β) + cos(α + β)] sin(α + β) = sin α cos β + cos α sin β sin(α − β) = sin α cos β − cos α sin β cos(α + β) = cos α cos β + sin α sin β cos(α − β) = cos α cos β − sin α sin β sin(2θ) = 2 sin θ cos θ sin^2 θ + cos^2 θ = 1
sinn^ x dx = −
n sinn−^1 x cos x + n − 1 n
sinn−^2 x dx
∫ cosn^ x dx =
n cosn−^1 x sin x + n − 1 n
cosn−^2 x dx
MATH 142 – Final Exam
∣∣ ∣∣
∫ (^) b a
f (x)dx − Mn
∣∣ ∣∣ ≤ (b^ −^ a)
(^3) K 2 24 n^2
∣∣ ∣∣
∫ (^) b a
f (x)dx − Tn
∣∣ ∣∣ ≤ (b^ −^ a)
(^3) K 2 12 n^2
∣∣ ∣∣
∫ (^) b a
f (x)dx − Sn
∣∣ ∣∣ ≤ (b^ −^ a)
(^5) K 4 180(2n)^4
u dv = uv −
v du
∫ (^) b
a
f (x) dx = lim q→ b−
∫ (^) q
a
f (x) dx
du u^2 + a^2
a tan−^1 u a
+ C
du u^2 − a^2
2 a
ln
u − a u + a
∣ +^ C
du √ u^2 + b
= ln
∣∣u + √u (^2) + b
∣∣ + C
√ du a^2 − u^2
= sin−^1 u a
+ C
tan−^1 x =
∑^ ∞
k=
(−1)k 2 k + 1 x^2 k+1^ − 1 ≤ x ≤ 1
(1 + x)m^ = 1 +
∑^ ∞
k=
m(m − 1)...(m − k + 1) k! xk^ − 1 < x < 1 (m 6 = 0, 1 , 2 , ...)
r = r(θ) : dy dx
r cos θ + dr dθ sin θ
−r sin θ +
dr dθ cos θ
