List of top Mathematics Questions on Integration asked in CBSE CLASS XII, CBSE Compartment XII

A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is $3$ m as shown. Based on the given information answer the following:

(i) Express $y$ as a function of $x$ from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to $x$.
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points $P$ and $Q$ where the outer edge of the track cuts $x$-axis and $y$-axis in first quadrant and find the area of triangle formed by points $P,O,Q$.

Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.

Find : \[ I = \int \frac{x + \sin x}{1 + \cos x} \, dx \]

The integrating factor of the differential equation $ (x + 2y^3) \frac{dy}{dx} = 2y $ is:

If $ \int \frac{1}{2x^2} \, dx = k \cdot 2x + C $, then $ k $ is equal to:

Find: \[ \int \frac{\cos x \, dx}{1 + \cos x + \sin x}. \]

Evaluate: \[ \int_{\frac{\pi}{2}}^{\pi} \frac{e^{x} \left(1 - \sin x \right)}{1 - \cos x} \, dx. \]

If $ \int_0^1 \frac{e^x}{1+x} \, dx = \alpha $, then $ \int_0^1 \frac{e^x}{(1+x)^2} \, dx $ is equal to:

The value of \[ \int_0^1 \frac{dx}{e^x + e^{-x}} \] is :

A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top into 4 equal quadrants and discovered the scratch passing through the origin inclined at an angle $ \frac{\pi}{4} $ anticlockwise along the positive direction of x-axis. Find the area of the region enclosed by the x-axis, the scratch and the circular table top in the first quadrant, using integration.

Find: \[ \int \frac{\sin^{-1} \left( \frac{x}{\sqrt{a + x}} \right)}{ \sqrt{a + x}} \, dx \]

The integrating factor of the differential equation $ \frac{dy}{dx} + y = \frac{1 + y}{x} $ is:

$ \int \frac{e^{10 \log x} - e^{8 \log x}}{e^{6 \log x} - e^{5 \log x}} \, dx$ is equal to :

Draw a rough sketch of the curve $ y = \sqrt{x} $. Using integration, find the area of the region bounded by the curve $ y = \sqrt{x} $, $ x = 4 $, and the x-axis, in the first quadrant.