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Let \( I(x)=\displaystyle \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{5}{13}}} \). If \( I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}}\right) \), \( b,c\in \mathbb{N} \), then \( 3(b+c) \) is equal to:

If the area of region \( \{(x,y)\,:\, 1 + x^{2} \le y \le \min\{x+7,\, 11-3x\}\} \) is \( A \), then \( 3A \) equal to:

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

If \( \displaystyle \int e^x \left( \frac{x\sin^{-1}x}{\sqrt{1-x^2}} + \frac{\sin^{-1}x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C \), where \( C \) is the constant of integration, then \(\displaystyle g\left(\frac{1}{2}\right) \) equals:

Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration.
If \(\displaystyle 8\Big(g\!\left(\frac{\pi}{2}\right) + g'\!\left(\frac{\pi}{2}\right)\Big) = \alpha \pi^3 + \beta \pi^2 + \gamma, \, \alpha,\beta,\gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals:

\( \int \frac{\left(\sqrt{1+x^{2}}+x\right)^{10}}{\left(\sqrt{1+x^{2}}-x\right)^{9}} \, dx = \frac{1}{m}\left(\left(\sqrt{1+x^{2}}+x\right)^{n}\left[n\sqrt{1+x^{2}}-x\right]\right)+C \)
If where \( C \) is the constant of Integration and \( m , n \in \mathbb{N} \) then \( m+n \) is equal to ______________.

If \( \displaystyle \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt[23]{3x^{-24} + x^{-26}} \right) dx = -\frac{\alpha}{3(\alpha + 1)} \left( 3x^\beta + x^\gamma \right)^{\frac{\alpha + 1}{\alpha}} + C, \quad x > 0, \, (\alpha, \beta, \gamma \in \mathbb{Z}) \), where \( C \) is the constant of integration, then \( \alpha+\beta+\gamma \) is equal to __________.

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

The integral \( \int_{0}^{\pi} \frac{8x \, dx}{4\cos^{2} x + \sin^{2} x} \) is equal to:

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JEE Main 2025 7th April Evening Shift

If the area of region \( \{(x,y)\,:\, 1 + x^{2} \le y \le \min\{x+7,\, 11-3x\}\} \) is \( A \), then \( 3A \) equal to: