List of top Mathematics Questions on Integration asked in CUET (UG)

Water is leaking from the bottom of a conical funnel at the rate of \(0.15\pi\) cm3/s. If the radius of the base is 10 cm and the height is 20 cm, then the rate at which the water level is dropping when it is 5 cm from the top is: 

The integral of \( \int_{-2}^{2} x^4 \, dx \) denominator \( (1+5x^2) \) is:
If \( \int \frac{1 + \cos\theta}{\tan 2\theta - \cot 2\theta} d\theta = \lambda \cos\theta + c \), then \( \lambda \) is equal to (where c is constant of integration)
The integral \( \int \frac{\cos 5x + \cos 4x}{1 - 2\cos 3x} \, dx \) is equal to (where C is an arbitrary constant):
If $\int_{0}^{a}\sqrt{x}dx = \frac{4a}{3}$, then $\int_{a}^{a+1}x\,dx$ is:
Evaluate \[ \int \frac{1+x+\sqrt{x+x^{2}}}{\sqrt{1+x}+\sqrt{x}}dx \]
Evaluate: \[ \int \dfrac{x^2+1}{x^4+1}\,dx \]

Match the following integrals in Column I with their corresponding results in Column II: 


 

Column IColumn II
A. \[ \displaystyle \int \frac{dx}{x^2-a^2} \]I. \[ \displaystyle \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C \]
B. \[ \displaystyle \int \frac{dx}{a^2-x^2} \]II. \[ \displaystyle \log\left|x+\sqrt{x^2-a^2}\right|+C \]
C. \[ \displaystyle \int \frac{dx}{\sqrt{x^2-a^2}} \]III. \[ \displaystyle \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C \]
D. \[ \displaystyle \int \frac{dx}{\sqrt{a^2-x^2}} \]IV. \[ \displaystyle \sin^{-1}\frac{x}{a}+C \]
Evaluate: \[ \int_{0}^{1} \frac{1}{1+x^2}\,dx \]
Evaluate: \[ \int \frac{1}{x^2 + 4x + 5}\,dx \]
Evaluate: \( \int_0^1 (2x+1) dx \)