Class XII calculus formula sheet: derivatives and integrals
The standard derivatives, integrals and properties of definite integrals you will use in almost every Class XII calculus question.
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These are the standard results you will use constantly in Class XII calculus. Learn them until you can recall them instantly — but also make sure you understand where each one comes from, because board questions often test the idea behind the formula. Here, log means the natural logarithm, loge, as in NCERT.
Rules of differentiation
| Rule | Result |
|---|---|
| Power rule | d⁄dx (xn) = nxn−1 |
| Product rule | (uv)′ = u′v + uv′ |
| Quotient rule | (u/v)′ = (u′v − uv′) / v2, v ≠ 0 |
| Chain rule | d⁄dx f(g(x)) = f′(g(x)) · g′(x) |
Standard derivatives
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec2 x |
| cot x | −cosec2 x |
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
| ex | ex |
| ax | ax log a |
| log x (x > 0) | 1/x |
| sin−1 x | 1/√(1 − x2), −1 < x < 1 |
| cos−1 x | −1/√(1 − x2), −1 < x < 1 |
| tan−1 x | 1/(1 + x2) |
Standard integrals
Each result holds up to an arbitrary constant C.
| Integral | Result |
|---|---|
| ∫ xn dx | xn+1/(n + 1) + C, n ≠ −1 |
| ∫ (1/x) dx | log |x| + C |
| ∫ ex dx | ex + C |
| ∫ ax dx | ax/log a + C |
| ∫ sin x dx | −cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec2 x dx | tan x + C |
| ∫ cosec2 x dx | −cot x + C |
| ∫ sec x tan x dx | sec x + C |
| ∫ tan x dx | log |sec x| + C |
| ∫ sec x dx | log |sec x + tan x| + C |
| ∫ dx/(x2 + a2) | (1/a) tan−1(x/a) + C |
| ∫ dx/√(a2 − x2) | sin−1(x/a) + C |
| ∫ dx/(x2 − a2) | (1/2a) log |(x − a)/(x + a)| + C |
| ∫ dx/(a2 − x2) | (1/2a) log |(a + x)/(a − x)| + C |
| ∫ dx/√(x2 + a2) | log |x + √(x2 + a2)| + C |
| ∫ dx/√(x2 − a2) | log |x + √(x2 − a2)| + C |
Integration by parts
∫ f(x) g(x) dx = f(x) ∫ g(x) dx − ∫ [ f′(x) · ∫ g(x) dx ] dx
To choose which function to treat as f, the ILATE order is a useful guide: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. It is a guideline, not a rule — always check that the new integral is simpler than the one you started with.
A special case worth remembering: ∫ ex [f(x) + f′(x)] dx = ex f(x) + C.
Properties of definite integrals
- ∫ab f(x) dx = −∫ba f(x) dx
- ∫ab f(x) dx = ∫ab f(a + b − x) dx
- ∫0a f(x) dx = ∫0a f(a − x) dx
- ∫−aa f(x) dx = 2∫0a f(x) dx if f is even, and 0 if f is odd
Common mistakes with these formulas
- Forgetting + C in indefinite integrals.
- Dropping the modulus in log |x|.
- Applying the power rule to ∫ x−1 dx, where n = −1 is excluded.
- Missing the chain-rule factor: the derivative of sin 3x is 3 cos 3x, not cos 3x.
- Ignoring the domain restrictions on the inverse trigonometric derivatives.