2. Calculus — Integration

Properties of definite integrals

The properties of definite integrals form the analytical backbone of integral calculus, enabling rapid simplification and evaluation of complex area-under-curve problems. This topic covers symmetry, interval splitting, and shifting properties.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Fundamental Integration Properties & King's Property

$$\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$$ and for symmetric intervals, $$\int_{-a}^{a} f(x)\,dx = \begin{cases} 2 \int_{0}^{a} f(x)\,dx & \text{if } f(-x) = f(x) \\ 0 & \text{if } f(-x) = -f(x) \end{cases}$$

📐 Geometric Interpretation

Reflecting the region of integration horizontally across the midpoint of the interval $(a+b)/2$ leaves the net accumulated area invariant. For symmetric intervals, even functions double the positive-side area due to mirror symmetry about the y-axis, whereas odd functions cancel out perfectly across the origin.

⚙️ Algebraic & Structural Meaning

Converts intractable integrals into symmetrical counterparts whose sum eliminates complex polynomial or trigonometric denominators instantly through variable substitution.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Continuity on $[a, b]$	`$[a, b]$ (Closed interval)`	Strictly Mandatory / Necessary	Discontinuities (especially infinite or jump discontinuities) render standard Riemann integration properties invalid without improper integral limits.
Piecewise Continuous on Subdivisions	`$[a, c]$ and $[c, b]$`	Mandatory for Interval Additivity Property	Failure to check domain boundaries when splitting integrals across points of definition change leads to divergent or incorrect evaluations.
Symmetry Domain for Parity Rules	$[-a, a]$	Essential for Even/Odd Reduction	Applying even/odd shortcuts on asymmetric intervals like $[-a, b]$ where $a \neq b$ yields completely erroneous results.

Useful results and shortcuts

Use these identities and results when their conditions apply:

King's Property (Interval Summation)

$$\int_a^b \frac{f(x)}{f(x) + f(a+b-x)}\,dx = \frac{b-a}{2}$$

Rule: If the integrand takes the form $f(x)/(f(x) + f(a+b-x))$, the answer is always half the length of the interval, $(b-a)/2$, bypassing integration entirely.

Application: Extensively tested in UP PGT and KVS with trigonometric and logarithmic functions.

Symmetric Bounds Reduction

$$\int_{-a}^{a} \frac{x^2 \sin^3 x}{1 + \cos^2 x}\,dx = 0$$

Rule: Instantly check the parity of $f(-x)$. If $f(-x) = -f(x)$ over $[-a, a]$, immediately mark $0$ without calculation.

Periodic Function Property

$$\int_0^{nT} f(x)\,dx = n \int_0^T f(x)\,dx$$

Rule: For a periodic function with period $T$, integer multiples of the period scale the integral linearly.

Application: Standard recurring pattern with fractional part functions and modulated trig functions.

Integration by Shifting Bounds

$$\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$$

Rule: Replace every $x$ with $a+b-x$ and add the original integral to eliminate complex variable terms.

Application: Crucial for inverse trigonometric and logarithmic definite integrals.

Wallis's Formula for Definite Integrals

$$\int_0^{\pi/2} \sin^n x \cos^m x \, dx$$

Rule: Use the factorial reduction product formula when limits are $0$ to $\pi/2$ and powers are integers.

Application: High-frequency appearance in DSSSB and UP PGT calculus sections.

Leibniz Rule for Integral Differentiation

$$\frac{d}{dx} \int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))h'(x) - f(g(x))g'(x)$$

Rule: Differentiate under the integral sign with variable limits instantly by substituting limits and multiplying by their derivatives.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Blindly Applying Even/Odd Property on Non-Symmetric Intervals

Common Pitfall: Applying $\int_{-a}^a f(x)\,dx = 0$ when the lower limit is not the negative of the upper limit (e.g., $\int_{-1}^2 x^3 \, dx$).

Counterexample: $$\int_{-1}^{2} x^3 \, dx \neq 0$$

Why it fails: The symmetry property requires exact balance $[-a, a]$. Asymmetric intervals leave unbalanced remnants.

🛡️ Exam Defense: Always check if the lower and upper limits are exact opposites before checking function parity.

⚠️ COMMON ERROR #02

Ignoring Domain Restrictions in Logarithmic/Trigonometric Integrands

Common Pitfall: Using King's property or substitution when the integrand has undefined points (asymptotes) inside the interval of integration.

Counterexample: $$\int_{-1}^{1} \frac{1}{x^2}\,dx$$

Why it fails: The function blows up at $x=0$, violating continuity on $[-1, 1]$. Standard definite integral properties do not apply directly.

🛡️ Exam Defense: Check for vertical asymptotes or points of infinite discontinuity within $[a, b]$ before evaluating.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
King's Property	Continuity on $[a, b]$	$$\int_a^b f(x) = \int_a^b f(a+b-x)$$	Horizontal reflection preserving area sum	Denominators of form $f(x) + f(a+b-x)$
Symmetric Parity	Continuity on $[-a, a]$	$$\int_{-a}^a f(x) = 2\int_0^a f(x) \text{ (even)}$$	Area duplication or exact cancellation	Limits of form $[-a, a]$ with polynomial/trig mix
Periodic Property	Continuity & Period $T$	$$\int_0^{nT} f(x) = n\int_0^T f(x)$$	Repetitive tile area accumulation	Integrals involving $\sin x$, $\|\sin x\|$, or \{x\}
Leibniz Rule	Continuity & Differentiability	$$\frac{d}{dx} \int_{a}^{g(x)} f(t)\,dt = f(g(x))g'(x)$$	Rate of change of variable area region	Integral equations with variable upper limits

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Evaluate the definite integral: $$\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx$$

(A) $\frac{\pi}{4}$(B) $\frac{\pi}{2}$(C) $\pi$(D) $0$

📚 Method A: Formal Textbook Derivation

Step 1: Let $I = \int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx$. Step 2: Apply property $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$ to get $I = \int_0^{\pi/2} \frac{\cos x}{\cos x + \sin x}\,dx$. Step 3: Add both equations to get $2I = \int_0^{\pi/2} 1\,dx = \pi/2$. Step 4: Thus $I = \pi/4$.

⚡ Method B: 25-Second Exam Shortcut

Recognize King's Property standard template where $f(x) + f(a+b-x)$ matches the denominator. Answer is instantly $(b-a)/2 = (\pi/2 - 0)/2 = \pi/4$.

💡 Tip: Memorize this standard result; examiners frequently change $\sin x$ to $\tan x$ or $\sec x$ with identical outcomes.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Evaluate: $$\int_{-3}^{3} x \sqrt{9-x^2}\,dx$$

(A) $18$(B) $9\pi$(C) $0$(D) $36$

📚 Method A: Formal Textbook Derivation

Step 1: Identify interval as $[-3, 3]$. Step 2: Let $f(x) = x \sqrt{9-x^2}$. Step 3: Test parity: $f(-x) = (-x)\sqrt{9-(-x)^2} = -x\sqrt{9-x^2} = -f(x)$. Step 4: Since $f(x)$ is an odd function on a symmetric interval, the integral is $0$.

⚡ Method B: 25-Second Exam Shortcut

Spot odd power of $x$ multiplied by even function on symmetric limits $[-a, a]$. Mark $0$ in 5 seconds.

💡 Tip: Always check for odd functions when limits are symmetric (e.g., $-a$ to $a$).

Practice questions

Choose an answer to check it and see the solution:

The value of $\int_{-\pi/2}^{\pi/2} \sin^7 x \cos^2 x \, dx$ is:

A $\frac{2}{63}$

B $0$

C $\frac{\pi}{4}$

D $1$

Derivation & Analysis:

The integrand $f(x) = \sin^7 x \cos^2 x$ satisfies $f(-x) = \sin^7(-x)\cos^2(-x) = -\sin^7 x \cos^2 x = -f(x)$. Thus, by the odd function property on a symmetric interval $[-\pi/2, \pi/2]$, the integral is $0$.

Evaluate $$\int_0^1 \log \left(\frac{1-x}{x}\right)\,dx$$

A $1$

B $\pi$

C $0$

D $-1$

Derivation & Analysis:

Let $I = \int_0^1 \log(\frac{1-x}{x})\,dx$. Using property $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$, we get $I = \int_0^1 \log(\frac{x}{1-x})\,dx = -I$. Therefore $2I = 0 \implies I = 0$.

What is the value of $\int_0^{\pi/2} \log(\tan x)\,dx$?

A $\pi$

B $1$

C $-\pi$

D $0$

Derivation & Analysis:

Using King's property, $\int_0^{\pi/2} \log(\tan x)\,dx = \int_0^{\pi/2} \log(\cot x)\,dx = -\int_0^{\pi/2} \log(\tan x)\,dx$. Thus $2I = 0 \implies I = 0$.

If $f(x)$ is continuous and $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$, then $\int_0^a x f(x)\,dx$ is equal to:

A $$\frac{a}{2} \int_0^a f(x)\,dx$$

B $$a \int_0^a f(x)\,dx$$

C $$0$$

D $$\int_0^a f(x)\,dx$$

Derivation & Analysis:

Using the property $\int_0^a x f(x)\,dx = \frac{a}{2} \int_0^a f(x)\,dx$, which is derived by replacing $x$ with $a-x$ and adding.

The value of $\int_{-1}^{1} |x|\,dx$ is:

A $0$

B $1$

C $2$

D $\frac{1}{2}$

Derivation & Analysis:

$|x|$ is an even function on $[-1, 1]$. Therefore, $\int_{-1}^1 |x|\,dx = 2 \int_0^1 x\,dx = 2 \left[\frac{x^2}{2}\right]_0^1 = 1$.

If $f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$, then $\int_{-1}^{1} f(x)\,dx$ depends only on:

A Odd powers of $x$

B Even powers of $x$

C All powers of $x$

D Constant term only

Derivation & Analysis:

On a symmetric interval $[-1, 1]$, odd powers integrate to $0$. Only even powers contribute $2 \int_0^1 a_{2k} x^{2k} \, dx$.

Evaluate $$\int_0^2 [x] \, dx$$ where $[x]$ is the greatest integer function:

A $0$

B $1$

C $2$

D $3$

Derivation & Analysis:

Split the interval: $\int_0^1 0\,dx + \int_1^2 1\,dx = 0 + 1(2-1) = 1$.

The derivative of $\int_0^{x^2} \cos(t^2)\,dt$ with respect to $x$ is:

A $\cos(x^2)$

B $2x \cos(x^4)$

C $x^2 \cos(x^2)$

D $2x \cos(x^2)$

Derivation & Analysis:

By Leibniz rule: $\frac{d}{dx} \int_0^{x^2} \cos(t^2)\,dt = \cos((x^2)^2) \cdot \frac{d}{dx}(x^2) = 2x \cos(x^4)$.

The value of $\int_{-\pi/2}^{\pi/2} \frac{x^2 \sin x}{1 + \cos^2 x}\,dx$ is:

A $\pi$

B $\frac{\pi^2}{2}$

C $0$

D $\frac{\pi}{2}$

Derivation & Analysis:

Let $f(x) = \frac{x^2 \sin x}{1 + \cos^2 x}$. Then $f(-x) = \frac{(-x)^2 \sin(-x)}{1 + \cos^2(-x)} = \frac{-x^2 \sin x}{1 + \cos^2 x} = -f(x)$. Since the integrand is odd and limits are symmetric, the integral is $0$.

Q10

If $f(x)$ is a periodic function with period $T$, then $\int_a^{a+T} f(x)\,dx$ is:

A Independent of $a$

B Dependent on $a$

C Always zero

D Equal to $T \int_0^T f(x)\,dx$

Derivation & Analysis:

A fundamental property of periodic functions states that $\int_a^{a+T} f(x)\,dx = \int_0^T f(x)\,dx$, making it independent of the starting point $a$.

Common questions

Yes! King's Property states $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$ for any general limits $a$ and $b$, though it is most frequently simplified when limits are $0$ to $a$.

Applying the even/odd property when the interval is not strictly symmetric (e.g., from $-1$ to $2$), or ignoring points of infinite discontinuity inside the integration interval.

Memorize the standard structural templates where denominators are sums of complementary terms like $\sin x + \cos x$ or $\sqrt{x} + \sqrt{a-x}$. The answer is always $(b-a)/2$.

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