Mathematics formulas

Subject Name: Applied Mathematics-I Faculty: T P Thrivikram

Algebraic Expressions xn+1
• ∫ x n ⁡dx = +c
n+1
•
d
(x n ) = nx n−1 • ∫ 0⁡dx = c
dx ax
•
d
(ax ) = ax log⁡(a) • ∫ ax ⁡dx = log⁡(a) + c
dx
d • ∫ ex ⁡dx = ex + c
• (ex ) = ex
dx eax
d 1 • ∫ eax ⁡dx = +c
• √x = 2√x a
dx
d⁡
• ∫ ⁡dx = x + c
• (Number) = 0 •
1
dx ∫ x ⁡dx = log(x) + c
d⁡ 1
• log(x) = x •
d
dx ∫ U. V⁡dx = U ∫ V⁡dx − ∫[dx U](∫ V⁡dx) ⁡dx + c
d
• f(g(x)) = f ′ (g(x)). g ′ (x) •
dx x
dx ∫ √a2 −x2 ⁡⁡⁡⁡ = sin−1 (a) + c
d dU dV
• (U. V) = V + U 1 1 x
dx dx dx • ∫ x2+a2 ⁡dx = a tan−1 (a) + c
dU dV
d U Vdx⁡⁡−⁡U dx dx 1 x
• ( )⁡= • ∫ x⁡√x2−a2 ⁡⁡⁡⁡ = a sec −1 (a) + c
dx V V2
dx
Trigonometric Expressions • ∫ x⁡√x2−1 ⁡⁡⁡⁡ = sec −1 (x) + c
dx 1 x−a
d • ∫ x2−a2 ⁡⁡⁡⁡ = 2a log (x+a) + c
• sin(x) = cos(x)
dx dx 1 a+x
d • ∫ a2 −x2 ⁡⁡⁡⁡ = 2a log (a−x) + c
• cos(x) = −⁡sin⁡(x)
dx −1 1 x
d • ∫ x⁡√x2−a2 dx ⁡⁡⁡⁡ = a cosec −1 (a) + c
• tan(x) = sec 2 (x)⁡
dx −1
d • ∫ x⁡√x2−1 dx ⁡⁡⁡⁡ = cosec −1 (x) + c
• cot(x) = −⁡cosec 2 (x)
dx
d • ∫ sin(x)⁡dx = − cos(x) + c
• sec(x) = sec(x) . tan⁡(x)
dx • ∫ cos(x)⁡dx = sin(x) + c
d
• cosec(x) = − cosec(x). cot⁡(x) • ∫ cosec 2 (x)⁡dx = −cot(x) + c
dx
• ∫ sec 2 (x) ⁡dx = tan(x) + c
Inverse Trigonometric Functions • ∫ sec(x). tan(x)⁡dx = sec(x) ⁡ + c
•
d
sin−1 (x) =
1 • ∫ cosec(x). cot(x)⁡dx = −⁡cosec(x) + c
√1−x2
dx
d −1
• ∫ cosec(x)⁡dx = −log[cosec(x) + cot(x)] + c
• −1
cos (x) = • ∫ sec(x)⁡dx = log[sec(x) + tan⁡(x)] + c
dx √1−x2
•
d
tan−1 (x) =
1
⁡ • ∫ tan(x)⁡dx = log[sec(x)] + c
dx 1+x2
d −1 • ∫ cot(x)⁡dx = log[sin(x)] + c
• 𝑐𝑜𝑡 −1 (x) = − cos(ax)
1+x2
dx
d 1 • ∫ sin(ax)⁡dx = +c
• sec −1 (x) = a
sin(ax)
dx |x|√x2−1
d −1
• ∫ cos(ax)⁡dx = a +c
• −1
cosec (x) = −cot(ax)
dx |x|√x2−1 • ∫ cosec 2 (ax)⁡dx = +c
a
tan(ax)
Hyperbolic Expressions • ∫ sec 2 (ax) ⁡dx = ⁡⁡ + c
a
sec(ax)
•
d
sinh(x) = cosh(x) • ∫ sec(ax). tan(ax)⁡dx = ⁡+c
dx a
d cosec(ax)
• cosh(x) = sinh(x) • ∫ cosec(ax). cot(ax)⁡dx = − +c
dx a
d −log⁡[cosec(ax)+cot(ax)]
• tan h(x) = sech2 (x)⁡ • ∫ cosec(ax)⁡dx = + c
dx a
d log[sec(ax)+tan⁡(ax)]
• cot h(x) = −⁡cosech2 (x) • ∫ sec(ax)⁡dx = a
+c
dx
d log[sec(ax)]
• sech(x) = −sec h(x) . tanh(x) • ∫ tan(ax)⁡dx = +c
dx a
d log[sin(ax)]
• cosech(x) = − cosec h(x). coth(x) • ∫ cot(ax)⁡dx = +c
dx a