JEE Main 2026 (Online) 23rd January Morning Shift | Indefinite Integrals Question 6 | Mathematics

Let $f(x)=\int \frac{\left(2-x^2\right) \cdot \mathrm{e}^x}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to:

$\sqrt{2 \mathrm{e}}-1$

$\sqrt{2 \mathrm{e}}+1$

$\sqrt{3 \mathrm{e}}-1$

$\sqrt{3 \mathrm{e}}+1$

JEE Main 2025 (Online) 3rd April Morning Shift

MCQ (Single Correct Answer)

-1

$$ \text { Let } f(x)=\int x^3 \sqrt{3-x^2} d x \text {. If } 5 f(\sqrt{2})=-4 \text {, then } f(1) \text { is equal to } $$

$-\frac{6 \sqrt{2}}{5}$

$-\frac{8 \sqrt{2}}{5}$

$-\frac{2 \sqrt{2}}{5}$

$-\frac{4 \sqrt{2}}{5}$

JEE Main 2025 (Online) 28th January Evening Shift

MCQ (Single Correct Answer)

-1

If $f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6$, then $f(1)$ is equal to :

$4\left(\log _{\mathrm{e}} 2-2\right)$

$\log _{e^2} 2+2$

$2-\log \mathrm{e}^2$

$4\left(\log _e 2+2\right)$

JEE Main 2025 (Online) 23rd January Evening Shift

MCQ (Single Correct Answer)

-1

Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z$, then $\alpha+\beta-\gamma$ equals :