List of top Mathematics Questions on Methods of Integration

\( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{dx}{1+\cot x} = \) _____

If \[ \int \frac{\tan x}{1+\tan x+\tan^2x}\,dx = x-\frac{K}{\sqrt A} \tan^{-1} \left( \frac{K\tan x+1}{\sqrt A} \right) +C, \] then the ordered pair \((K,A)\) is

Evaluate \[ \int \frac{dx}{(1+\sqrt{x})\sqrt{x-x^2}}. \]

Find: \[ \int \frac{\cos x}{(2+\sin x)(4+\sin x)}\,dx. \]

Find : \( \int \frac{dx}{x^{1/2} + x^{1/3}} \)

The indefinite integral $\int \frac{1}{1 + \cos x} \, dx$ is equal to:

$\int \frac{\sqrt{\sqrt{x} + 1}}{\sqrt{x}} dx = $}

If $\int \frac{5\sqrt{x}(4x^2 - 1)}{x} dx = 8\sqrt{x^5} + a\sqrt{x} + C$, where $a$ is a constant and $C$ is the constant of integration, then the value of $a$ is

\[ \int \left(27x^3(1-x^3)\right)^{\frac{2}{3}}\,dx = \]

\(\int \frac{\sec x \sqrt{\sec x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx = \)

Evaluate \[ \int\frac1{2\cot x-3\tan x}\,dx \]

If \[ \int(e^{2x}+2e^x)\sqrt{e^{2x}-4e^x+5}\,dx = \frac13[f(x)]^{3/2} + 4\left[ \frac{e^x-2}{2}\sqrt{f(x)} + \frac12g(x) \right]+c \] then \(f(0)=\)

The indefinite integral of \( \sin(x) \) w.r.t \( \cos(x) \) is:

Evaluate the integral: \[ \int \frac{x}{x+2}\, dx \]

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

$\int e^{x}[\frac{1}{1+x}-\frac{1}{(1+x)^{2}}]dx=$ ________.

$\int\frac{\cos \theta}{2-\sin^{2}\theta}d \theta=$ ________.

$\int\frac{\log(1+x)}{(1+x)}dx=$ ________.

If $\int \frac{2x+3}{(x-1)(x^2+1)} dx = \log_e \left\{ (x - 1)^{\frac{5}{2}} (x^2 + 1)^a \right\} - \frac{1}{2} \tan^{-1} x + A$ where A is an arbitrary constant, then the value of $a$ is

\(\int_0^2 \frac{3x+1}{x^2+4} dx\)

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: