List of top Mathematics Questions on Methods of Integration

The general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \) is _____
\( \int \frac{e^{2025x} + e^{-2025x}}{e^{2025x} + e^{-2025x}} dx = \)_____ + C
\( \int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =\) _____ + C
\( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{dx}{1+\cot x} = \) _____
Evaluate the integral \( \int \frac{x}{x^2 + 1} dx \):
The integral $ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx $ is:
The integral $ \int e^x \left( \frac{x + 5}{(x + 6)^2} \right) dx $ is:
The integral \[ \int \sqrt{1 + \sin x} \, dx \] is equal to:
The integral \[ \int_0^{\frac{\pi}{2}} \cos x \cdot e^{\sin x} \, dx \] is equal to:
The integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \] is:

The integral $ \int_0^1 \frac{1}{2 + \sqrt{2e}} \, dx $ is:

Sketch a graph of \( y = x^2 \). Using integration, find the area of the region bounded by \( y = 9 \), \( x = 0 \), and \( y = x^2 \).

Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]

Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
Evaluate the integral \( \int x e^{x^2} dx \):
Evaluate the integral  \(\int_{0}^{1} \frac{1}{\sqrt{3+x} + \sqrt{1+x}} \, dx\) Given that the integral can be expressed in the form \(a + b\sqrt{2} + c\sqrt{3}\), where \(a, b, c\) are rational numbers, find the value of \(2a + 3b - 4c\).

If $\phi(x)=\frac{1}{\sqrt{ x }} \int_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) dt , x>$, then $\emptyset^{\prime}\left(\frac{\pi}{4}\right)$ is equal to :

The integral \(\int_0^{\frac{\pi}{2}} \frac{1}{3 + 2\sin x + \cos x} \, dx\) is equal to
The value of $\int \limits \frac {dx}{x^2(x^4+1)^{3/4}}$ is
The integral $\int^{^{\frac{1}{2}}}_{_0} \frac{ln \left(1+2x\right)}{1+4x^{2}}dx,$ equals :