PYQs Guides Notes Mocks Board Questions NDA Mathematics Definite Integration 69 NDA Mathematics previous year questions (PYQs) from 19 papers, 2017 to 2026, with answers and worked solutions.
About this chapter Difficulty: 19 easy · 37 moderate · 13 hard. Most-asked subtopics: Properties of Definite Integrals — Symmetry, King's, Odd/Even (33), Integration of Absolute Value, Piecewise, and Greatest Integer Functions (17), Fundamental Theorem, Periodic Integrals, and Leibniz Rule (13). Updated 14 September 2026. #1 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ∫ 0 1 d x ( a x + b x + c ) 2 \int_0^1 \frac{dx}{(ax + bx + c)^2} ∫ 0 1 ( a x + b x + c ) 2 d x equal to? A − 1 a + b + c -\frac{1}{a + b + c} − a + b + c 1 B − 1 c ( a + b + c ) -\frac{1}{c(a + b + c)} − c ( a + b + c ) 1 C 1 a + b + c \frac{1}{a + b + c} a + b + c 1 D 1 c ( a + b + c ) \frac{1}{c(a + b + c)} c ( a + b + c ) 1 #2 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ∫ 1 2 x x ( 1 + ln x ) d x \int_1^2 x^x (1 + \ln x)\,dx ∫ 1 2 x x ( 1 + ln x ) d x equal to? #3 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
If ∫ a b x 3 d x = 0 \int_a^b x^3\,dx=0 ∫ a b x 3 d x = 0 and ∫ a b x 2 d x = 2 / 3 \int_a^b x^2\,dx=2/3 ∫ a b x 2 d x = 2/3 , then a and b are respectively? #4 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ∫ − 1 1 { d d x ( tan − 1 1 x ) } d x \int_{-1}^{1}\left\{\frac{d}{dx}\left(\tan^{-1}\frac{1}{x}\right)\right\}dx ∫ − 1 1 { d x d ( tan − 1 x 1 ) } d x equal to? #5 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
If f ( x ) f(x) f ( x ) satisfies f ( 1 ) = f ( 4 ) f(1)=f(4) f ( 1 ) = f ( 4 ) , then what is ∫ 1 4 f ′ ( x ) d x \int_{1}^{4}f'(x)\,dx ∫ 1 4 f ′ ( x ) d x equal to? Done a few? Here’s your next move
#6 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ∫ 0 π / 2 e ln ( cos x ) d x \int_{0}^{\pi/2}e^{\ln(\cos x)}\,dx ∫ 0 π /2 e l n ( c o s x ) d x equal to? #7 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
If f ( x ) = 2 x f(x)=2^{x} f ( x ) = 2 x , then what is ∫ 2 10 f ′ ( x ) f ( x ) d x \int_{2}^{10}\frac{f'(x)}{f(x)}\,dx ∫ 2 10 f ( x ) f ′ ( x ) d x equal to? #8 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ∫ 0 π 4 d x ( sin x + cos x ) 2 \int_0^{\frac{\pi}{4}}\frac{dx}{(\sin x+\cos x)^2} ∫ 0 4 π ( s i n x + c o s x ) 2 d x equal to? #9 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
If ∫ 0 π / 2 ( sin 4 x + cos 4 x ) d x = k \int_0^{\pi/2}(\sin^4x+\cos^4x)\,dx = k ∫ 0 π /2 ( sin 4 x + cos 4 x ) d x = k , then what is the value of ∫ 0 20 π ( sin 4 x + cos 4 x ) d x \int_0^{20\pi}(\sin^4x+\cos^4x)\,dx ∫ 0 20 π ( sin 4 x + cos 4 x ) d x ? #10 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
If I 1 = ∫ e e 2 d x ln x I_1=\int_e^{e^2}\dfrac{dx}{\ln x} I 1 = ∫ e e 2 ln x d x and I 2 = ∫ 1 2 e x x d x I_2=\int_1^2\dfrac{e^x}{x}\,dx I 2 = ∫ 1 2 x e x d x , then which one of the following is correct? A I 1 − I 2 = 0 I_1-I_2=0 I 1 − I 2 = 0 B I 1 + I 2 = 0 I_1+I_2=0 I 1 + I 2 = 0 C I 1 − 2 I 2 = 0 I_1-2I_2=0 I 1 − 2 I 2 = 0 D 2 I 1 − I 2 = 0 2I_1-I_2=0 2 I 1 − I 2 = 0 #11 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ∫ 0 8 π ∣ sin x ∣ d x \int_0^{8\pi}|\sin x|\,dx ∫ 0 8 π ∣ sin x ∣ d x equal to? Let ϕ ( a ) = ∫ a a + 100 π ∣ sin x ∣ d x \phi(a)=\displaystyle\int_a^{a+100\pi}|\sin x|\,dx ϕ ( a ) = ∫ a a + 100 π ∣ sin x ∣ d x . #12 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ϕ ( a ) \phi(a) ϕ ( a ) equal to? #13 · Fundamental Theorem, Periodic Integrals, and Leibniz Rule
What is ϕ ′ ( a ) \phi'(a) ϕ ′ ( a ) equal to? #14 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
If f ( x + y ) = f ( x ) + f ( y ) f(x + y) = f(x) + f(y) f ( x + y ) = f ( x ) + f ( y ) , then what is
∫ − 1 1 f ( x ) d x \int_{-1}^{1} f(x)\,dx ∫ − 1 1 f ( x ) d x
equal to? A B ∫ 0 1 f ( x ) d x \int_0^1 f(x)\,dx ∫ 0 1 f ( x ) d x C 2 ∫ 0 1 f ( x ) d x 2\int_0^1 f(x)\,dx 2 ∫ 0 1 f ( x ) d x D #15 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π / 2 d θ 1 + cos θ \int_0^{\pi/2}\dfrac{d\theta}{1+\cos\theta} ∫ 0 π /2 1 + cos θ d θ equal to? #16 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
If f ( x ) f(x) f ( x ) and g ( x ) g(x) g ( x ) are continuous functions satisfying f ( x ) = f ( a − x ) f(x)=f(a-x) f ( x ) = f ( a − x ) and g ( x ) + g ( a − x ) = 2 g(x)+g(a-x)=2 g ( x ) + g ( a − x ) = 2 , then what is ∫ 0 a f ( x ) g ( x ) d x \int_0^a f(x)g(x)\,dx ∫ 0 a f ( x ) g ( x ) d x equal to? A ∫ 0 a g ( x ) d x \int_0^a g(x)\,dx ∫ 0 a g ( x ) d x B ∫ 0 a f ( x ) d x \int_0^a f(x)\,dx ∫ 0 a f ( x ) d x C 2 ∫ 0 a f ( x ) d x 2\int_0^a f(x)\,dx 2 ∫ 0 a f ( x ) d x D #17 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 2 π 1 + sin x 2 d x \displaystyle\int_0^{2\pi} \sqrt{1 + \sin\dfrac{x}{2}}\,dx ∫ 0 2 π 1 + sin 2 x d x equal to? #18 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
The value of ∫ 0 π / 4 tan x d x + ∫ 0 π / 4 cot x d x \displaystyle\int_0^{\pi/4}\sqrt{\tan x}\,dx + \int_0^{\pi/4}\sqrt{\cot x}\,dx ∫ 0 π /4 tan x d x + ∫ 0 π /4 cot x d x is equal to A B C π 2 2 \dfrac{\pi}{2\sqrt{2}} 2 2 π D π 2 \dfrac{\pi}{\sqrt{2}} 2 π #19 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π e x sin x d x \int_{0}^{\pi}e^x\sin x\,dx ∫ 0 π e x sin x d x equal to? A e π + 1 2 \frac{e^\pi+1}{2} 2 e π + 1 B e π − 1 2 \frac{e^\pi-1}{2} 2 e π − 1 C D e π + 1 4 \frac{e^\pi+1}{4} 4 e π + 1 #20 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 1 e x ln x d x \int_{1}^{e}x\ln x\,dx ∫ 1 e x ln x d x equal to? A B e 2 + 1 4 \frac{e^2+1}{4} 4 e 2 + 1 C D e 2 − 1 4 \frac{e^2-1}{4} 4 e 2 − 1 #21 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ − π / 4 π / 4 ( sin x − tan x ) d x \int_{-\pi/4}^{\pi/4}(\sin x-\tan x)\,dx ∫ − π /4 π /4 ( sin x − tan x ) d x ? A − 1 2 + ln ( 1 2 ) -\frac{1}{\sqrt{2}}+\ln\!\left(\frac{1}{\sqrt{2}}\right) − 2 1 + ln ( 2 1 ) B C D #22 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 1 x ( 1 − x ) 9 d x \int_{0}^{1}x(1-x)^9\,dx ∫ 0 1 x ( 1 − x ) 9 d x equal to? #23 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
∫ 0 π / 2 e sin x cos x d x \displaystyle\int_0^{\pi/2}e^{\sin x}\cos x\,dx ∫ 0 π /2 e s i n x cos x d x is equal to Directions for the following two (02) items :
Read the following information and answer the two items that follow :
Consider the integrals I 1 = ∫ 0 π x d x 1 + sin x I_{1} = \displaystyle\int_{0}^{\pi} \dfrac{x\,dx}{1+\sin x} I 1 = ∫ 0 π 1 + sin x x d x and I 2 = ∫ 0 π ( π − x ) d x 1 − sin ( π + x ) I_{2} = \displaystyle\int_{0}^{\pi} \dfrac{(\pi-x)\,dx}{1-\sin(\pi+x)} I 2 = ∫ 0 π 1 − sin ( π + x ) ( π − x ) d x . #24 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of I 1 I_{1} I 1 ? #25 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of I 1 + I 2 I_{1} + I_{2} I 1 + I 2 ? #26 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of ∫ 0 π / 4 ( tan 3 x + tan x ) d x \int_0^{\pi/4}(\tan^3 x+\tan x)\,dx ∫ 0 π /4 ( tan 3 x + tan x ) d x ? #27 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 a f ( a − x ) f ( x ) + f ( a − x ) d x \int_0^a\frac{f(a-x)}{f(x)+f(a-x)}\,dx ∫ 0 a f ( x ) + f ( a − x ) f ( a − x ) d x equal to? #28 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
If ∫ 0 a [ f ( x ) + f ( − x ) ] d x = ∫ − a a g ( x ) d x \int_0^a[f(x)+f(-x)]\,dx=\int_{-a}^a g(x)\,dx ∫ 0 a [ f ( x ) + f ( − x )] d x = ∫ − a a g ( x ) d x , then what is g ( x ) g(x) g ( x ) equal to? A B f ( − x ) + f ( x ) f(-x)+f(x) f ( − x ) + f ( x ) C D #29 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π ln ( tan x 2 ) d x \int_{0}^{\pi}\ln\!\left(\tan\frac{x}{2}\right)dx ∫ 0 π ln ( tan 2 x ) d x equal to? #30 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 1 ln ( 1 x − 1 ) d x \int_0^1 \ln\!\left(\frac{1}{x}-1\right)dx ∫ 0 1 ln ( x 1 − 1 ) d x equal to? #31 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ − π / 2 π / 2 ( e cos x sin x + e sin x cos x ) d x \int_{-\pi/2}^{\pi/2}(e^{\cos x}\sin x+e^{\sin x}\cos x)\,dx ∫ − π /2 π /2 ( e c o s x sin x + e s i n x cos x ) d x equal to? A e 2 − 1 e \frac{e^2-1}{e} e e 2 − 1 B e 2 + 1 e \frac{e^2+1}{e} e e 2 + 1 C 1 − e 2 e \frac{1-e^2}{e} e 1 − e 2 D Consider the following for the next three (03) items that follow: Let I 1 = ∫ 0 π x 1 + cos 2 x d x I_{1}=\int_{0}^{\pi}\dfrac{x}{1+\cos^{2}x}\,dx I 1 = ∫ 0 π 1 + cos 2 x x d x and I 2 = ∫ 0 π 1 1 + sin 2 x d x I_{2}=\int_{0}^{\pi}\dfrac{1}{1+\sin^{2}x}\,dx I 2 = ∫ 0 π 1 + sin 2 x 1 d x . #32 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of I 1 + I 2 I 1 − I 2 \dfrac{I_{1}+I_{2}}{I_{1}-I_{2}} I 1 − I 2 I 1 + I 2 ? A B C D π + 1 π − 1 \dfrac{\pi+1}{\pi-1} π − 1 π + 1 #33 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of 8 I 1 2 8I_{1}^{2} 8 I 1 2 ? #34 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of I 2 I_{2} I 2 ? A π 2 \dfrac{\pi}{\sqrt{2}} 2 π B π 2 2 \dfrac{\pi}{2\sqrt{2}} 2 2 π C 3 π 2 2 \dfrac{3\pi}{2\sqrt{2}} 2 2 3 π D π 4 2 \dfrac{\pi}{4\sqrt{2}} 4 2 π For the next two (02) items that follow: Let ∫ 0 π / 2 a sin x + b cos x ( a + b ) ( sin x + cos x ) d x = k \int_0^{\pi/2}\dfrac{a\sin x+b\cos x}{(a+b)(\sin x+\cos x)}dx=k ∫ 0 π /2 ( a + b ) ( sin x + cos x ) a sin x + b cos x d x = k . #35 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is the value of k?
#36 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π / 2 a cos x + b sin x sin x + cos x d x \int_0^{\pi/2}\dfrac{a\cos x+b\sin x}{\sin x+\cos x}dx ∫ 0 π /2 sin x + cos x a cos x + b sin x d x equal to? For the next two (02) items that follow: Let f ( x ) = sin x f(x)=\sin x f ( x ) = sin x and g ( x ) − f ( x ) = f ( 4 − x ) g(x)-f(x)=f(4-x) g ( x ) − f ( x ) = f ( 4 − x ) . #37 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 4 f ( x ) g ( x ) d x \int_0^4\dfrac{f(x)}{g(x)}dx ∫ 0 4 g ( x ) f ( x ) d x equal to? #38 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 4 f ( 4 − x ) g ( 4 − x ) d x \int_0^4\dfrac{f(4-x)}{g(4-x)}dx ∫ 0 4 g ( 4 − x ) f ( 4 − x ) d x equal to? Let I = ∫ − 2 π 2 π sin 4 x + cos 4 x 1 + 3 x d x I=\int_{-2\pi}^{2\pi}\frac{\sin^4x+\cos^4x}{1+3^x}\,dx I = ∫ − 2 π 2 π 1 + 3 x s i n 4 x + c o s 4 x d x . #39 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π ( sin 4 x + cos 4 x ) d x \int_0^\pi(\sin^4x+\cos^4x)\,dx ∫ 0 π ( sin 4 x + cos 4 x ) d x equal to? #40 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
#41 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ − 1 1 ( 3 sin x − sin 3 x ) cos 2 x d x \displaystyle\int_{-1}^{1}(3\sin x-\sin3x)\cos^2x\,dx ∫ − 1 1 ( 3 sin x − sin 3 x ) cos 2 x d x equal to? #42 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π / 2 a + sin x 2 ( a + sin x + cos x ) d x \displaystyle\int_0^{\pi/2}\dfrac{a+\sin x}{2(a+\sin x+\cos x)}\,dx ∫ 0 π /2 2 ( a + sin x + cos x ) a + sin x d x equal to? Let 3 f ( x ) + f ( 1 x ) = 1 x + 1 3f(x)+f\!\left(\dfrac{1}{x}\right)=\dfrac{1}{x}+1 3 f ( x ) + f ( x 1 ) = x 1 + 1 . #43 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is 8 ∫ 1 2 f ( x ) d x 8\displaystyle\int_{1}^{2}f(x)\,dx 8 ∫ 1 2 f ( x ) d x equal to? A ln ( 8 e ) \ln(8\sqrt{e}) ln ( 8 e ) B C D Let f ( t ) = ln ( t + 1 + t 2 ) f(t)=\ln\!\left(t+\sqrt{1+t^2}\right) f ( t ) = ln ( t + 1 + t 2 ) and g ( t ) = tan ( f ( t ) ) g(t)=\tan(f(t)) g ( t ) = tan ( f ( t )) . #44 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ − π π g ( t ) d t \displaystyle\int_{-\pi}^{\pi}g(t)\,dt ∫ − π π g ( t ) d t equal to? Let I = ∫ 0 π / 2 f ( x ) g ( x ) d x I=\displaystyle\int_0^{\pi/2}f(x)g(x)\,dx I = ∫ 0 π /2 f ( x ) g ( x ) d x , where f ( x ) = sin x f(x)=\sin x f ( x ) = sin x and g ( x ) = sin x + cos x + 1 g(x)=\sin x+\cos x+1 g ( x ) = sin x + cos x + 1 . #45 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
What is ∫ 0 π / 2 d x g ( x ) \displaystyle\int_0^{\pi/2}\dfrac{dx}{g(x)} ∫ 0 π /2 g ( x ) d x equal to? A ln 2 2 \dfrac{\ln 2}{2} 2 ln 2 B ln 2 4 \dfrac{\ln 2}{4} 4 ln 2 C D #46 · Properties of Definite Integrals — Symmetry, King's, Odd/Even
A π 4 + ln 2 \dfrac{\pi}{4}+\ln 2 4 π + ln 2 B π 4 − ln 2 \dfrac{\pi}{4}-\ln 2 4 π − ln 2 C π 4 − ln 2 2 \dfrac{\pi}{4}-\dfrac{\ln 2}{2} 4 π − 2 ln 2 D π 4 + ln 2 2 \dfrac{\pi}{4}+\dfrac{\ln 2}{2} 4 π + 2 ln 2 #47 · Integration of Absolute Value, Piecewise, and Greatest Integer Functions
What is ∫ e − 1 e 2 ∣ ln x x ∣ d x \int_{e^{-1}}^{e^2}\left|\dfrac{\ln x}{x}\right|\,dx ∫ e − 1 e 2 x ln x d x equal to? #48 · Integration of Absolute Value, Piecewise, and Greatest Integer Functions
What is ∫ 0 2 [ x 2 ] d x \int_{0}^{\sqrt{2}}[x^2]\,dx ∫ 0 2 [ x 2 ] d x equal to (where [.] is greatest integer function)? A B C 2 ( 2 − 1 ) 2(\sqrt{2}-1) 2 ( 2 − 1 ) D #49 · Integration of Absolute Value, Piecewise, and Greatest Integer Functions
What is ∫ a b [ x ] d x + ∫ a b [ − x ] d x \int_a^b [x]\,dx + \int_a^b [-x]\,dx ∫ a b [ x ] d x + ∫ a b [ − x ] d x equal to, where [ . ] [.] [ . ] is the greatest integer function? #50 · Integration of Absolute Value, Piecewise, and Greatest Integer Functions
What is ∫ 2 8 ∣ x − 5 ∣ d x \int_2^8 |x-5|\,dx ∫ 2 8 ∣ x − 5∣ d x equal to? Showing the first 50 of 69.
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