Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202622 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
22 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2026 · 22 January · Morning Shift · Mathematics · Indefinite Integration

If ∫( x)^ (-11)/(2) ( x)^ (-5)/(2) d x= - p_ 1 q_ 1 ( x)^ (9)/(2) -...

If ∫(sin⁡x)−112(cos⁡x)−52dx=\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x= −p1q1(cot⁡x)92−p2q2(cot⁡x)52−p3q3(cot⁡x)12+p4q4(cot⁡x)−32+C-\frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}}-\frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}}-\frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}}+\frac{p_{4}}{q_{4}}(\cot x)^{\frac{-3}{2}}+\mathrm{C}, where piandp_{i} andq_{i}arepositiveintegerswithare positive integers with\operatorname{gcd}(pi,qi)=1\left(p_{i}, q_{i}\right)=1 for i=1,2,3,4i=1,2,3,4 and C is the constant of integration, then 15p1p2p3p4q1q2q3q4\frac{15 p_{1} p_{2} p_{3} p_{4}}{q_{1} q_{2} q_{3} q_{4}} is equal to

Correct answer

0

Step-by-step solution

∫sin⁡−11/2xcos⁡−5/2x dx\int \sin^{-11/2}x\cos^{-5/2}x\,dx.
Since m+n=−8m+n = -8 (negative even), substitute t=cot⁡xt = \cot x, dx=−dt1+t2dx = \frac{-dt}{1+t^2}.
Integral becomes −∫(1+t2)3t5/2dt=−∫(t−5/2+3t−1/2+3t3/2+t7/2)dt-\int \frac{(1+t^2)^3}{t^{5/2}}dt = -\int(t^{-5/2}+3t^{-1/2}+3t^{3/2}+t^{7/2})dt.
=23t−3/2−6t1/2−65t5/2−29t9/2+C= \frac{2}{3}t^{-3/2} - 6t^{1/2} - \frac{6}{5}t^{5/2} - \frac{2}{9}t^{9/2} + C.
Comparing: p1q1=29\frac{p_1}{q_1} = \frac{2}{9}, p2q2=65\frac{p_2}{q_2} = \frac{6}{5}, p3q3=61\frac{p_3}{q_3} = \frac{6}{1}, p4q4=23\frac{p_4}{q_4} = \frac{2}{3}.
15p1p2p3p4q1q2q3q4=15⋅2⋅6⋅6⋅29⋅5⋅1⋅3=2160135=16\frac{15p_1p_2p_3p_4}{q_1q_2q_3q_4} = \frac{15 \cdot 2 \cdot 6 \cdot 6 \cdot 2}{9 \cdot 5 \cdot 1 \cdot 3} = \frac{2160}{135} = 16.
Practice Indefinite Integration on Quantrex Academy →

More from Indefinite Integration

Let f(x) = ∫ ((16x + 24)/(x^2 + 2x - 15)) dx . If f(4) = 14 _e(3) and f(7) = _e(2^ α · 3^… JEE Main 2026Let f(x)=∫ d x x^ ((2)/(3)) +2 x^ ((1)/(2)) be such that f(0)=-26+24 _ e (2) . If… JEE Main 2026Let f(t)=∫( 1- ( _ e t) 1- ( _ e t) ) d t, t>1 . If f(e^ π / 2 )=-e^ π / 2 and f(e^ π / 4… JEE Main 2026Let I(x)=∫ 3 d x (4 x+6)( 4 x^ 2 +8 x+3 ) and I(0)= 3 4 +20 . If I((1)/(2))= a 2 b +c… JEE Main 2026Let f(x)=∫ (2-x^ 2 ) · e^ x ( 1+x )(1-x)^ 3 / 2 ~d x . If f(0)=0 , then f((1)/(2)) is… JEE Main 2026If aligned ∫((1)/(x)+(1)/(x^3)) & ( [23] 3 x^ -24 +x^ -26 ) d x & =-(α)/(3(α+1))(3… JEE Main 2025If ∫ ( 1+x^2 +x)^ 10 ( 1+x^2 -x)^9 d x= (1)/(m)(( 1+x^2 +x)^n(n 1+x^2 -x))+C where C is… JEE Main 2025Let f(x)=∫ x^3 3-x^2 d x . If 5 f( 2 )=-4 , then f(1) is equal to JEE Main 2025If f(x)=∫ 1 x^ 1 / 4 (1+x^ 1 / 4 ) d x, f(0)=-6 , then f(1) is equal to : JEE Main 2025If ∫ 2 x^2+5 x+9 x^2+x+1 ~d x=x x^2+x+1 +α x^2+x+1 +β _e|x+(1)/(2)+ x^2+x+1 |+C , where C… JEE Main 2025Let ∫ x^3 x ~d x=g(x)+C , where C is the constant of integration. If 8(g((π)/(2))+g^… JEE Main 2025Let I(x)=∫ d x (x-11)^ (11)/(13) (x+15)^ (15)/(13) . If I(37)-I(24)=(1)/(4)( 1 ~b^… JEE Main 2025If ∫ e^x( x ^ -1 x 1-x^2 + ^ -1 x (1-x^2)^ 3 / 2 +(x)/(1-x^2)) d x=g(x)+C , where C is… JEE Main 2025Let ∫ (2- x)/(3+ x) ~d x=(1)/(2)(α x+ _ e |β x+γ x|)+C , where C is the constant of… JEE Main 2024If ∫ 1 [5] (x-1)^4(x+3)^6 ~d x=A((α x-1)/(β x+3))^B+C , where C is the constant of… JEE Main 2024Let I(x)=∫ (6)/( ^2 x(1- x)^2) d x . If I(0)=3 , then I((π)/(12)) is equal to JEE Main 2024 Full Indefinite Integration list Indefinite Integration JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs