Indefinite Integration

7 JEE Main questions · Maths

MathsJEE Main 22 Jan 2026, Shift 1·Q23·Numerical·Moderate
Open
Integration by substitution
If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}}\,dx = -\frac{p_1}{q_1}(\cot x)^{\frac92} - \frac{p_2}{q_2}(\cot x)^{\frac52} - \frac{p_3}{q_3}(\cot x)^{\frac12} + \frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}} + C,$ where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i) = 1$ for $i = 1, 2, 3, 4$ and $C$ is the constant of integration, then $\dfrac{15p_1p_2p_3p_4}{q_1q_2q_3q_4}$ is equal to .
MathsJEE Main 23 Jan 2026, Shift 1·Q2·MCQ·Moderate
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Integral of type e^x(f+f')
Let $f(x) = \displaystyle\int\frac{(2 - x^2)\cdot e^x}{\left(\sqrt{1+x}\right)(1-x)^{3/2}}\,dx$ . If $f(0) = 0$ , then $f\left(\dfrac12\right)$ is equal to:
1. (1) $\sqrt{3e} - 1$
2. (2) $\sqrt{2e} + 1$
3. (3) $\sqrt{2e} - 1$
4. (4) $\sqrt{3e} + 1$
We corrected a printing error in the original paper
Source printed “If $f(0) - 0$ ” (minus sign instead of the equals sign).
MathsJEE Main 23 Jan 2026, Shift 2·Q14·MCQ·Moderate
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Integration by substitution
Let $I(x) = \displaystyle\int\frac{3\,dx}{(4x + 6)\left(\sqrt{4x^2 + 8x + 3}\right)}$ and $I(0) = \dfrac{\sqrt3}{4} + 20$ . If $I\left(\dfrac12\right) = \dfrac{a\sqrt2}{b} + c$ , where $a, b, c \in \mathbb{N}$ , $\gcd(a, b) = 1$ , then $a + b + c$ is equal to:
1. (1)29
2. (2)28
3. (3)31
4. (4)30
MathsJEE Main 24 Jan 2026, Shift 1·Q11·MCQ·Moderate
Open
Special integral of exponential form
Let $f(t) = \displaystyle\int\left(\dfrac{1 - \sin(\log_e t)}{1 - \cos(\log_e t)}\right)dt$ , $t > 1$ . If $f\left(e^{\pi/2}\right) = -e^{\pi/2}$ and $f\left(e^{\pi/4}\right) = \alpha\, e^{\pi/4}$ , then $\alpha$ equals
1. (1) $-1 - \sqrt2$
2. (2) $-1 - 2\sqrt2$
3. (3) $1 + \sqrt2$
4. (4) $-1 + \sqrt2$
MathsJEE Main 28 Jan 2026, Shift 1·Q11·MCQ·Moderate
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Integration by parts
If $\displaystyle\int\left(\frac{1 - 5\cos^2 x}{\sin^5 x\cos^2 x}\right)dx = f(x) + C$ where $C$ is the constant of integration, then $f\left(\dfrac{\pi}{6}\right) - f\left(\dfrac{\pi}{4}\right)$ is equal to
1. (1) $\dfrac{1}{\sqrt3}\left(26 + \sqrt3\right)$
2. (2) $\dfrac{4}{\sqrt3}\left(8 - \sqrt6\right)$
3. (3) $\dfrac{1}{\sqrt3}\left(26 - \sqrt3\right)$
4. (4) $\dfrac{2}{\sqrt3}\left(4 + \sqrt6\right)$
MathsJEE Main 28 Jan 2026, Shift 2·Q19·MCQ·Moderate
Open
Integration by substitution
Let $f(x) = \displaystyle\int\frac{dx}{x^{\left(\frac23\right)} + 2x^{\left(\frac12\right)}}$ be such that $f(0) = -26 + 24\log_e(2)$ . If $f(1) = a + b\log_e(3)$ , where $a, b \in \mathbb{Z}$ , then $a + b$ is equal to:
1. (1) $-18$
2. (2) $-5$
3. (3) $-11$
4. (4) $-26$
MathsJEE Main 2 Apr 2026, Shift 2·Q19·MCQ·Easy–moderate
Open
Integration by partial fractions
Let $f(x) = \displaystyle\int\left(\frac{16x + 24}{x^2 + 2x - 15}\right)dx$ . If $f(4) = 14\log_e(3)$ and $f(7) = \log_e(2^{\alpha}\cdot3^{\beta})$ , $\alpha, \beta \in \mathbb{N}$ , then $\alpha + \beta$ is equal to:
1. (1)31
2. (2)37
3. (3)39
4. (4)41