“ILATE”
Choosing u in integration by parts
Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. Whichever comes first in that list becomes u.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Integrals and Their Applications formulas
Integration is differentiation read backwards, and the marks come from recognising the shape of the integrand. The standard forms and the two methods, substitution and by parts, cover nearly every question set.
19 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Standard integrals
| What | Formula | Symbols and notes |
|---|---|---|
| Power rule | ∫xn dx = xn+1/(n + 1) + c | for n ≠ −1 |
| Reciprocal | ∫(1/x) dx = ln|x| + c | the exception to the power rule |
| Exponential | ∫ex dx = ex + c | and ∫ax dx = ax/ln a + c |
| Trigonometric | ∫sin x dx = −cos x + c, ∫cos x dx = sin x + c | the minus sits with sine |
| Secant squared | ∫sec²x dx = tan x + c | and ∫cosec²x dx = −cot x + c |
| Inverse forms | ∫dx/(1 + x²) = tan−1x + c | and ∫dx/√(1 − x²) = sin−1x + c |
| Difference of squares | ∫dx/(x² − a²) = (1/2a)ln|(x − a)/(x + a)| + c | partial fractions in disguise |
Methods
| What | Formula | Symbols and notes |
|---|---|---|
| Substitution | put u = g(x), then du = g'(x)dx | look for a function and its derivative together |
| Integration by parts | ∫u dv = uv − ∫v du | choose u by ILATE |
| Special by-parts result | ∫ex(f(x) + f'(x))dx = exf(x) + c | spot the pattern and skip the work |
| Partial fractions | split a rational function into simpler ones | then integrate term by term |
| Trigonometric powers | use identities to lower the power first | for example cos²x = (1 + cos2x)/2 |
Definite integrals and area
| What | Formula | Symbols and notes |
|---|---|---|
| Fundamental theorem | ∫ from a to b of f(x)dx = F(b) − F(a) | no constant of integration |
| Reversal | ∫ from a to b = −∫ from b to a | swapping the limits changes the sign |
| King's property | ∫ from 0 to a of f(x)dx = ∫ from 0 to a of f(a − x)dx | the most used property in exams |
| Even function | ∫ from −a to a of f(x)dx = 2∫ from 0 to a | when f(−x) = f(x) |
| Odd function | ∫ from −a to a of f(x)dx = 0 | when f(−x) = −f(x) |
| Area under a curve | A = ∫ y dx between the limits | take the modulus where the curve dips below the axis |
| Area between two curves | A = ∫(upper − lower)dx | find the intersection points first |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Odd over a symmetric range is zero”
Definite integrals
If f(−x) = −f(x) and the limits are −a to a, the answer is zero without any work. Check the symmetry before integrating.
“Substitution wants a pair”
Spotting when to substitute
Look for a function and its own derivative in the same integrand. That pairing is the signal to substitute.
“Definite needs no c”
Constants of integration
The constant cancels between the two limits, so a definite integral never carries + c.
Where marks go
The mistakes that cost the most.
- Using the power rule for 1/x, which needs the logarithm instead.
- Forgetting + c on an indefinite integral, a mark lost for nothing.
- Changing the variable in a substitution without changing the limits in a definite integral.
- Taking area as the plain integral when the curve crosses the x axis; the parts below count as positive area.
- Choosing u by habit rather than by ILATE, which turns a short problem into a long one.
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