Complete Integral Calculus: Formulas and Techniques

Integral Calculus Reference 1



The Complete Integral Calculus Reference

1 Basic Integral Formulas 1.4 Inverse Trigonometric
1.1 Power Rule Z
dx x
• √ = sin−1 + C
Z
xn+1 −xa2
2 a
• xn dx = + C (n ̸= −1) Z
dx 1 x
n+1 • = tan−1 + C
Z 2
a +x 2 a a
1
• dx = ln |x| + C Z
dx 1 x
x • √ = sec−1 + C
2
x x −a 2 a a
1.2 Exponential & Logarithmic
Z 1.5 Hyperbolic
• ex dx = ex + C Z
• sinh x dx = cosh x + C
ax
Z
• x
a dx = +C Z
ln a
Z • cosh x dx = sinh x + C
• ln x dx = x ln x − x + C
1.6 Rational (Denominator Forms)
1.3 Trigonometric Z
dx 1 a+x
• = ln +C
a2 2
Z
−x 2a a−x
• sin x dx = − cos x + C
x−a
Z
dx 1
Z • 2 2
= ln +C
• cos x dx = sin x + C x −a 2a x+a
Z
dx p
Z • √ = ln x + x2 ± a2 + C
• sec2 x dx = tan x + C x2 ± a 2
Z 1.7 Square-Root (Algebraic) Forms
• csc2 x dx = − cot x + C
a2
Z p
xp 2 x
Z • a2 − x2 dx = a − x2 + sin−1 + C
2 2 a
• sec x tan x dx = sec x + C Z p
Z • x2 + a2 dx =
• csc x cot x dx = − csc x + C
xp 2 a2 p
Z x + a2 + ln x + x2 + a2 + C
• tan x dx = ln | sec x| + C Z2 p 2
Z • x2 − a2 dx =
• cot x dx = ln | sin x| + C xp 2 a2 p
x − a2 − ln x + x2 − a2 + C
Z 2 2
• sec x dx = ln | sec x + tan x| + C
1.8 General Rules
Z Z Z Z
• csc x dx = ln | csc x − cot x| + C • [f ± g] dx = f dx ± g dx
Z Z
• kf (x) dx = k f (x) dx
Z
1
• f (ax+b) dx = F (ax+b) + C
a


2 Integration by Parts
Z Z
• u dv = uv − v du

2.1 Choosing u: The ILATE Rule
Priority order for u — whichever type appears first becomes u; the rest becomes dv.

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