int_formulas_fin.pdf - Mathematics 115 Basic Integral...

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Unformatted text preview: Mathematics 115 Basic Integral Formulas and Trig Identities II Integral Formulas Z Z xn+1 x dx = + C, n 6= −1 n+1 Z 1 dx = ln |x| + C x Z sin x dx = − cos x + C Z cos x dx = sin x + C Z sec 2 x dx = tan x + C Z csc2 x dx = − cot x + C Z sec x tan x dx = sec x + C Z csc x cot x dx = − csc x + C Z cot x dx = ln | sin x| + C Z 1 cos ax dx = sin ax + C a Z 1 cosh ax dx = sinh ax + C a Z sec x dx = ln |sec x + tan x| + C Z Z u dv = uv − v du Z 1 x dx = ln(x2 + a2 ) + C 2 2 x +a 2 n 1 dx = tan−1 x + C 1 + x2 1 √ dx = sin−1 x + C = − cos−1 x + C 2 1−x 1 √ dx = sec −1 x + C x x2 − 1 1 x 1 dx = tan−1 + C 2 2 x +a a a 1 x √ dx = sin−1 + C a a2 − x2 Z Z Z Z Z Z ex dx = ex + C 1 ax e +C a Z ax ax dx = +C ln a Z tan x dx = − ln | cos x| + C Z 1 sin ax dx = − cos ax + C a Z 1 sinh ax dx = cosh ax + C a Z csc x dx = ln | csc x − cot x| + C Z 1 1 dx = ln |ax + b| + C ax + b a eax dx = Trigonometric Identities 1 ; sin x 1 ; cos x • csc x = • sin2 x + cos2 x = 1; • sin(−x) = − sin x; • sin(x ± y) = sin x cos y ± cos x sin y; • sin 2x = 2 sin x cos x; • cos2 x = 21 (1 + cos 2x); • cos a cos b = 12 [cos(a − b) + cos(a + b)]; sec x = cot x = 1 ; tan x tan2 x + 1 = sec 2 x; tan x = sin x ; cos x cot x = cos x sin x 1 + cot2 x = csc2 x cos(−x) = cos x cos(x ± y) = cos x cos y ∓ sin x sin y cos 2x = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x sin2 x = 21 (1 − cos 2x) sin a cos b = 21 [sin(a − b) + sin(a + b)] sin a sin b = 12 [cos(a − b) − cos(a + b)]; ...
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