List of top Mathematics Questions on Integration by Parts asked in JEE Main

Let ∫x3sin⁡x dx=g(x)+C,where C is the constant of integration. \int x^3 \sin x \, dx = g(x) + C, \quad \text{where \( C \) is the constant of integration.} If g(π2)+g(π2)=απ3+βπ2+γ,α,β,γ∈Z, g\left( \frac{\pi}{2} \right) + g\left( \frac{\pi}{2} \right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \quad \alpha, \beta, \gamma \in {Z}, then α+β−γ equals: \alpha + \beta - \gamma \text{ equals:}
Let f(x)=∫0x2t2−8t+15etdt, x∈R f(x) = \int_0^{x^2 \frac{t^2 - 8t + 15}{e^t}} dt, \, x \in \mathbb{R} . Then the numbers of local maximum and local minimum points of f f , respectively, are:
If ∫(xsin⁡−1x+sin⁡−1x(1−x2)3/2+x1−x2)dx=g(x)+C \int \left( x \sin^{-1} x + \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 g(12) g\left(\frac{1}{2}\right) equals:
If f(x)+f(π−x)=π2f(x) + f(\pi-x) = \pi^{2} then 0∫πf(x)sinxdx_{0}\int^{\pi}f(x) sinx dx ?
If ∫01(x21+x14+x7)(2x14+3x7+6)1/7dx=1l(11)m/n\int\limits_0^1\left(x^{21}+x^{14}+x^7\right)\left(2 x^{14}+3 x^7+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n} where l,m,n∈N,ml, m, n \in N , m and nn are coprime then l+m+nl+m+n is equal to ______

In(x)=∫0x1(t2+5)ndt,n=1,2,3,⋯\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}

Then

If dydx+2ytan⁡x=sin⁡x,0\frac{d y}{d x}+2 y \tan x=\sin x, 0

If ∫dxa2sin⁡x+b2cos⁡2x=112tan⁡−1(3tan⁡x)+c,\int\frac{dx}{a^2\sin x+b^2\cos^2x}=\frac{1}{12}\tan^{-1}(3\tan x)+c,then the maximum value of a sinx + bcosx is____
Find ∫−π2π2sin⁡2x1+2x  dx\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \frac{\sin^2x}{1+2^x} \;dx