List of top Mathematics Questions on Methods of Integration

\( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{dx}{1+\cot x} = \) _____
\( \int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =\) _____ + C
\( \int \frac{e^{2025x} + e^{-2025x}}{e^{2025x} + e^{-2025x}} dx = \)_____ + C
The general solution of the differential equation \( \frac{dy}{dx} = e^{x+y} \) is _____
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 integrating factor of the differential equation \[ e^{-\frac{2x}{\sqrt{x}}} \, \frac{dy}{dx} = 1 \] 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 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 \(2a + 3b - 4c\).

Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \] 

Evaluate the integral: \[ I = \int_{-3}^{3} |2 - x| dx. \] 

If \[ \int \frac{3}{2\cos 3x \sqrt{2} \sin 2x} dx = \frac{3}{2} (\tan x)^{\beta} + \frac{3}{10} (\tan x)^4 + C \] then \( A = \) ? 

Evaluate the integral: \[ I = \int \frac{\cos x + x \sin x}{x (x + \cos x)} dx =\] 

Find the area of the region (in square units) enclosed by the curves: \[ y^2 = 8(x+2), \quad y^2 = 4(1-x) \] and the Y-axis. 

Evaluate the integral: $$ I = \int_{\frac{1}{\sqrt[5]{32}}}^{\frac{1}{\sqrt[5]{31}}} \frac{1}{\sqrt[5]{x^{30} + x^{25}}} \, dx. $$ 
Evaluate the integral: \[ \int \frac{4\tan^4 x + 3 \tan^2 x - 1}{\tan^2 x + 4} \, dx. \]
If \( \int x^3 \sin 3x \,dx = f(x) \cos 3x + g(x) \sin 3x + c \), then evaluate \( 27(f(x) + xg(x)) \):
Evaluate the integral: \[ \int_0^{16} \frac{\sqrt{x}}{1 + \sqrt{x}} dx. \]