Is it correct to say: Substituting u = g(x) but keeping the old x-limits?

Change the limits to g(a) and g(b) in the same line as the substitution, or return to x before applying the limits — never mix the two.

Is it correct to say: Using f(a − x) when the lower limit is not zero?

∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx needs lower limit 0. For limits a to b use f(a + b − x).

Is it correct to say: Calling an integrand odd or even after looking at one factor?

Compute f(−x) for the whole integrand. x³ + x² is neither; split it into its odd part x³ and even part x² and treat each separately.

Is it correct to say: Integrating |x − c| or a piecewise function in one go?

Split the interval at c, write the modulus without the bars on each piece (with the correct sign), then add.

Is it correct to say: Applying a property and stopping, without adding the two forms of I?

The trick works because I (original) + I (after the property) simplifies. Write both, add them to get 2I, then divide.

Is it correct to say: Taking h = 1/n in a limit-of-sum question when b − a is not 1?

Use h = (b − a)/n and, as in the textbook, the general term h·f(a + rh) for r = 1 to n. Take the limit only after simplifying with the Σ formulas.

📄 Formula sheet (PDF)

Definite Integration

Definite integral as a limit of a sum, the fundamental theorem of integral calculus, properties of definite integrals and methods of evaluation — Maharashtra HSC Maths Part II Ch 4

📐 15 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Scan the limits before you integrate: 0 to a, a to b, −a to a and 0 to 2a each point to a different property.
→In property questions write I = …, apply the property to get a second form of I, then add the two forms. Show both forms clearly — that is where the method lies.
→For limit-of-sum questions write h = (b − a)/n and the general term h·f(a + rh) first, and quote Σr, Σr², Σr³ before substituting.
→Proofs of the standard properties can be asked. Each proof is one substitution (x = a − t, x = a + b − t or x = −t) followed by a change of limits.
→After a substitution, change the limits at once. Never put x-limits on an integral written in u.
→Sanity check: if f(x) ≥ 0 on [a, b] with a < b, the value of the integral cannot be negative.

📐 Formulas(15)

Definite Integral as Limit of a Sum★ Board fav

\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{r = 1}^{n} h f (a + r h), h = \frac{b - a}{n}

Symbol	Meaning
$a, b$	Lower and upper limits of integration
$n$	Number of equal sub-intervals
$h$	Width of each sub-interval, (b − a)/n

Standard Sums for Limit-of-Sum Questions

\sum_{r = 1}^{n} r = \frac{n ( n + 1 )}{2}, \sum_{r = 1}^{n} r^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}, \sum_{r = 1}^{n} r^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Fundamental Theorem of Integral Calculus★ Board fav

\int_{a}^{b} f (x) d x = [F (x)]_{a}^{b} = F (b) - F (a), F^{'} (x) = f (x)

Symbol	Meaning
$F (x)$	Any antiderivative (indefinite integral) of f(x)

Property: Change of Variable Name

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (t) d t

Property: Interchanging the Limits

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, \int_{a}^{a} f (x) d x = 0

Property: Splitting the Interval

\int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x, a < c < b

Property: a + b − x★ Board fav

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

Property: a − x (lower limit zero)★ Board fav

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

Property: Splitting 0 to 2a

\int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x

Property: 0 to 2a with f(2a − x) = ± f(x)

\int_{0}^{2 a} f (x) d x = {2 \int_{0}^{a} f (x) d x, 0, f (2 a - x) = f (x) f (2 a - x) = - f (x)

Property: Even and Odd Functions★ Board fav

\int_{- a}^{a} f (x) d x = {2 \int_{0}^{a} f (x) d x, 0, f (- x) = f (x) (even) f (- x) = - f (x) (odd)

Substitution in a Definite Integral

\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u, u = g (x)

Integration by Parts for Definite Integrals

\int_{a}^{b} u v d x = [u \int v d x]_{a}^{b} - \int_{a}^{b} [\frac{d u}{d x} \int v d x] d x

Standard Result: sinⁿ x over sinⁿ x + cosⁿ x

\int_{0}^{π /2} \frac{s i n ^{n} x}{s i n ^{n} x + c o s ^{n} x} d x = \int_{0}^{π /2} \frac{c o s ^{n} x}{s i n ^{n} x + c o s ^{n} x} d x = \frac{π}{4}

Standard Result: ∫ log sin x from 0 to π/2

\int_{0}^{π /2} lo g sin x d x = \int_{0}^{π /2} lo g cos x d x = - \frac{π}{2} lo g 2

✏️ Solved Examples

1Solved Exampleeasy3 steps

Evaluate ∫₁² (3x² − 2x + 1) dx.

Find an antiderivative term by term.

F (x) = x^{3} - x^{2} + x

2Solved Exampleboard5 steps

Evaluate ∫₀² x² dx as a limit of a sum.

Here a = 0, b = 2, so h = 2/n and f(a + rh) = (rh)² = r²h².

h = \frac{2}{n}, f (a + r h) = r^{2} h^{2}

3Solved Exampleboard4 steps

Evaluate ∫₀^{π/2} √(sin x) / (√(sin x) + √(cos x)) dx.

Call the integral I.

I = \int_{0}^{π /2} \frac{s i n x}{s i n x + c o s x} d x

4Solved ExampleHOTS4 steps

Evaluate ∫₀^{π/4} log(1 + tan x) dx.

Let I be the integral and apply the a − x property with a = π/4.

I = \int_{0}^{π /4} lo g [1 + tan (\frac{π}{4} - x)] d x

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Substituting u = g(x) but keeping the old x-limits
✓Change the limits to g(a) and g(b) in the same line as the substitution, or return to x before applying the limits — never mix the two.
2
Using f(a − x) when the lower limit is not zero
✓∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx needs lower limit 0. For limits a to b use f(a + b − x).
3
Calling an integrand odd or even after looking at one factor
✓Compute f(−x) for the whole integrand. x³ + x² is neither; split it into its odd part x³ and even part x² and treat each separately.
4
Integrating |x − c| or a piecewise function in one go
✓Split the interval at c, write the modulus without the bars on each piece (with the correct sign), then add.
5
Applying a property and stopping, without adding the two forms of I
✓The trick works because I (original) + I (after the property) simplifies. Write both, add them to get 2I, then divide.
6
Taking h = 1/n in a limit-of-sum question when b − a is not 1
✓Use h = (b − a)/n and, as in the textbook, the general term h·f(a + rh) for r = 1 to n. Take the limit only after simplifying with the Σ formulas.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Evaluate ∫₀^{π/2} cos² x dx.
Q2
Evaluate ∫ from −π/4 to π/4 of x³ sin⁴ x dx.
Q3
Evaluate ∫₀³ |x − 1| dx.
Q4
Evaluate ∫₀¹ x(1 − x)⁵ dx using the a − x property.
Q5
Evaluate ∫₂⁵ √x / (√x + √(7 − x)) dx.
Q6
Evaluate ∫₁ᵉ log x dx.

📝 Notes

Definite Integration

This chapter turns the antiderivatives of indefinite integration into numbers. Most questions are either a direct evaluation using the fundamental theorem, an evaluation as a limit of a sum, or a "use a property" question where direct integration would be hard.

Three ways to evaluate

Fundamental theorem: find F with F′ = f, then compute F(b) − F(a). This is the default.
Limit of a sum: only when the question asks for it. Write $h = \frac{b-a}{n}$ , form $\sum h\,f(a + rh)$ , simplify with $\sum r$ , $\sum r^2$ , $\sum r^3$ (or a G.P. sum for $e^x$ ), and then let $n \to \infty$ .
Properties: when the integrand looks impossible but has a symmetry.

Choosing the right property

Look at the limits first.

0 to a — try $f(a - x)$ . With $a = \frac{\pi}{2}$ , sin and cos swap; with $a = \frac{\pi}{4}$ , $\tan\left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x}$ .
a to b — try $f(a + b - x)$ .
−a to a — test whether the integrand is even or odd. An odd integrand gives 0 immediately.
0 to 2a — compare $f(2a - x)$ with $f(x)$ .
Modulus or piecewise — split the interval at the break point.

Writing a property proof

Every proof follows the same pattern: substitute ( $x = a - t$ , $x = a + b - t$ or $x = -t$ ), write $dx$ in terms of $dt$ , change both limits, use the reversal property to fix the order of the limits, and finally rename $t$ as $x$ . Write each of these steps on its own line.

After applying a property

Write the original I and the transformed I one below the other, add them, and simplify the sum before integrating. If the sum does not simplify, try a different property.