Integral Calculus
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Standard Indefinite Integrals
Power Rule
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C
(n ≠ -1)
(n ≠ -1)
Logarithmic Rule
∫ (1/x) dx = ln|x| + C
Exponential Rules
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ / ln(a) + C
∫ aˣ dx = aˣ / ln(a) + C
Basic Trigonometric
∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ cos(x) dx = sin(x) + C
Secant/Cosecant Squared
∫ sec²(x) dx = tan(x) + C
∫ csc²(x) dx = -cot(x) + C
∫ csc²(x) dx = -cot(x) + C
Other Trig Products
∫ sec(x)tan(x) dx = sec(x) + C
∫ csc(x)cot(x) dx = -csc(x) + C
∫ csc(x)cot(x) dx = -csc(x) + C
Special Integrals & Inverse Trig Forms
Inverse Tan Form
∫ dx / (x² + a²) =
(1/a) tan⁻¹(x/a) + C
(1/a) tan⁻¹(x/a) + C
Inverse Sine Form
∫ dx / √(a² - x²) =
sin⁻¹(x/a) + C
sin⁻¹(x/a) + C
Logarithmic Forms
∫ dx / (x² - a²) =
(1/2a) ln|(x-a)/(x+a)| + C
(1/2a) ln|(x-a)/(x+a)| + C
Other Log Forms
∫ dx / √(x² ± a²) =
ln|x + √(x² ± a²)| + C
ln|x + √(x² ± a²)| + C
Core Trig Integrals
∫ tan(x) dx = ln|sec(x)| + C
∫ cot(x) dx = ln|sin(x)| + C
∫ cot(x) dx = ln|sin(x)| + C
Secant / Cosecant
∫ sec(x) dx = ln|sec(x)+tan(x)|
∫ csc(x) dx = ln|csc(x)-cot(x)|
∫ csc(x) dx = ln|csc(x)-cot(x)|
Methods of Integration
Integration by Parts (ILATE)
∫ u v dx = u ∫v dx - ∫(u' ∫v dx) dx
Choose 'u' using ILATE priority: Inverse Log, Algebraic, Trig, Exponential. 'v' is the other part.
Integration by Substitution
∫ f(g(x)) g'(x) dx
Let t = g(x), then dt = g'(x)dx
Becomes: ∫ f(t) dt
Let t = g(x), then dt = g'(x)dx
Becomes: ∫ f(t) dt
Look for a function and its derivative pair.
Partial Fractions (Linear)
For P(x) / (x-a)(x-b):
Decompose to A/(x-a) + B/(x-b) and integrate term-by-term.
Decompose to A/(x-a) + B/(x-b) and integrate term-by-term.
Special Form: eˣ [f(x) + f'(x)]
∫ eˣ [f(x) + f'(x)] dx = eˣ f(x) + C
A very common BITSAT shortcut. Identify f(x) and its derivative f'(x) quickly.
Definite Integrals & Properties
Fundamental Theorem (FTC)
∫ₐᵇ f(x) dx = F(b) - F(a)
where F'(x) = f(x).
where F'(x) = f(x).
King's Property (Most Important)
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx
Use this to simplify complex integrands, especially in trigonometry.
Even-Odd Function Property
∫₋ₐᵃ f(x) dx =
2 ∫₀ᵃ f(x) dx, if f(x) is even
0, if f(x) is odd
2 ∫₀ᵃ f(x) dx, if f(x) is even
0, if f(x) is odd
Queen's Property
∫₀²ᵃ f(x) dx =
2 ∫₀ᵃ f(x) dx, if f(2a-x) = f(x)
0, if f(2a-x) = -f(x)
2 ∫₀ᵃ f(x) dx, if f(2a-x) = f(x)
0, if f(2a-x) = -f(x)
BITSAT Integration Strategy
King's Property (∫f(a+b-x)) solves over 50% of tough definite integral problems. Always try it first.
For ∫eˣ(f(x)+f'(x))dx, quickly identify f(x) and f'(x). Common pairs: (ln x, 1/x), (tan⁻¹x, 1/(1+x²)).
See limits like -a to a? Immediately check if the function is even or odd. It's a massive time-saver.
Don't waste time on complex partial fractions. BITSAT questions use simple linear or repeated factors.
Memorize the results for ∫√(a²-x²), ∫√(x²+a²), etc. They are often asked directly.
Area Under Curves
Area with x-axis
Area = ∫ₐᵇ |y| dx = ∫ₐᵇ |f(x)| dx
Use absolute value. If f(x) is below the x-axis, the integral is negative, so area is -∫f(x)dx.
Area with y-axis
Area = ∫cᵈ |x| dy = ∫cᵈ |g(y)| dy
Use when the curve is defined as x in terms of y.
Area Between Two Curves
Area = ∫ₐᵇ |f(x) - g(x)| dx
Integrate (Top Curve - Bottom Curve) over the interval defined by their intersection points.
Indefinite vs. Definite Integrals
| Aspect | Indefinite Integral | Definite Integral |
|---|---|---|
Result | A family of functions (f(x) + C) | A single numerical value |
Constant 'C' | Always present | Not present (cancels out) |
Limits | No limits of integration | Has upper and lower limits |
Represents | Antiderivative of a function | Net signed area under the curve |