List of top Mathematics Questions on Integral Calculus

Let \( \int \frac{2 - \tan x}{3 + \tan x} \, dx = \frac{1}{2} \left( \alpha x + \log_e \lvert \beta \sin x + \gamma \cos x \rvert \right) + C \), where \( C \) is the constant of integration.

If \[ \int \sin(101x)\,\sin^{99}x\,dx = \frac{\sin(100x)\sin^{100}x}{k+5}+c, \] then \(\dfrac{k}{19}=\)

The integral \[ \int\!\frac{\sin^2x\cos^2x}{(\sin^5x+\cos^3x\sin^2x+\sin^3x\cos^2x+x^5+x\cos^5x)}dx \] is of the form}

The value of $ \int_0^{2\pi} \sqrt{1 + \sin^2 \frac{x}{2}} \, dx \text{ is} $

The value of the integral
\[\int_{0}^{\sqrt{8}} \int_{x^2}^{\sqrt{8 - x^2}} \frac{3\sqrt{x^2 + y^2}}{\sqrt{8}} \, dy \, dx \]
is equal to __________ (answer in integer).

Let \( f(x) = \int_{2-2x}^{x^2-1} e^{t^2-t} \, dt \) for all \( x \in \mathbb{R} \). If \( f \) is decreasing on \( (0, m) \) and increasing on \( (m, \infty) \), then the value of \( m \) is equal to __________ (answer in integer).

Consider the improper integrals
\[I_1 = \int_{1}^{\infty} \frac{t \sin t}{e^t} \, dt \]
and 
\[I_2 = \int_{1}^{\infty} \frac{1}{\sqrt{t}} \ln\left(1 + \frac{1}{t}\right) \, dt.\]
Then:

The number of integral solutions of \( \log_2 \left( \frac{x - 7}{2x - 3} \right) \geq 0 \) is:

Which of the following is/are TRUE ?

Let 𝛼 and 𝛽 be real constants such that\(lim_{x-0^+}\frac{∫^x_0(\frac{αt^2}{1+t^4})dt}{βx-sin\,x}\)=1 .
Then, the value of 𝛼 + 𝛽 equals ________

If, for some 𝛼∈(0, ∞),
\(∫^∞_02^{-x^2}dx=a\sqrt\pi,\) 
then 𝛼 equals (round off to two decimal places)

Evaluate the integral \[ \int \sqrt{\csc x - 1} \, dx = \]

Value of \(\int_c\frac{2z+1}{z^2+z}dz\), where C is\(|z|=\frac{1}{2}\), is:

If \( \int f(x)\,dx = g(x) \), then \( \int x^9 f(x^5)\,dx \) is equal to