List of top Mathematics Questions on Calculus asked in TS EAMCET

If \( \int x^3 \sin(3x) dx = \frac{1}{27} [f(x)\cos(3x) + g(x)\sin(3x)] + c \) then f(1)+g(1)=

The differential equation corresponding to the family of ellipses \( \frac{x^2}{a^2} + \frac{y^2}{4} = 1 \), where 'a' is an arbitrary constant is

If y=f(x) is the solution of the differential equation \( (1+\cos^2 x)f'(x) - 4\sin(2x) - f(x)\sin(2x) = 0 \) when f(0)=0, then \( f(\pi/3) = \)

\( \int_0^{\pi/2} \sqrt{\tan x} \, dx = \)

\( \int_{0}^{\pi/2} \frac{dx}{\cos x - \sqrt{3}\sin x} = \)

\( \int_{-1}^{5} \frac{1}{\sqrt{20+x-x^2}} dx = \)

\( \int \frac{1}{(x+2)\sqrt{x^2+x+2}} dx = \)

\( \int \frac{x+\cos x}{1-\sin x} dx = \)

If \( I_1 = \int \sin^6 x \, dx \) and \( I_2 = \int \cos^6 x \, dx \) then \( I_1 + I_2 = \)

If x and y are two positive real numbers such that xy=4 then the minimum value of \( \sqrt{x+\frac{y^2}{2}} \) is

There is a possible error of 0.02 cm in measuring the base diameter of a right circular cone as 14 cm. If the semi-vertical angle of the cone is 45°, then the approximate error in its volume is (in cu. cm)

A rod of length 41 m with an end A on the floor and another end B on the wall perpendicular to the floor is sliding away horizontally from the wall at the rate of 3 ft/min. When the end B is at the height of 9 ft from the floor, then the rate at which the area of the triangle formed by the rod with wall and floor changes at that instant is (in ft/min)

If the curves \(y^2=12x-3\) and \(y^2=12-kx\) cut each other orthogonally then the length of the sub tangent at (1,b) on the curve \(y^2=12-kx\) is

If \( y = x^{\log x} + (\log x)^x, x>1 \) then \( (\frac{dy}{dx})_{x=e} = \)

If \( y = \tan^2(\cos^{-1}\sqrt{\frac{1+x^2}{2}}) \), then \( \frac{dy}{dx} = \)

The domain of the derivative of the function \( f(x) = \cos^{-1}(2x-5) - \sin^{-1}(x-2) \) is

If \( x = 2\sqrt{2}\sqrt{\cos 2\theta} \) and \( y = 2\sqrt{2}\sqrt{\sin 2\theta} \), \( 0<\theta<\frac{\pi}{4} \) then the value of \( \frac{dy}{dx} \) at \( \theta = 22\frac{1}{2}^\circ \) is

If [x] is the greatest integer function and \( f(x) = \begin{cases} \frac{2[x]-x}{|x|} & x \neq 0 \\ 1 & x=0 \end{cases} \) is a real valued function, then f is

If \( \lim_{x \to 0} \frac{3x^3 - (1-x^2)^{3/2}}{x^2\sin x} = p + \log q \) then pq =

The equation which represents the system of parabolas whose axis is parallel to y-axis satisfies the differential equation

The substitution required to reduce the differential equation \(t^2 dx + (x^2 - tx + t^2) dt = 0\) to a differential equation which can be solved by variables separable method is

The area of the region bounded by the curves \(y=x^3\), \(y=x^2\) and the lines \(x=0\) and \(x=2\) is

Let m, n, p, q be four positive integers. If \(\int_0^{2\pi} \sin^m x \cos^n x dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x dx\), \(\int_0^{2\pi} \sin^p x \cos^q x dx = 0\), \(a = m+n+p\) and \(b = m+n+q\), then

\(\lim_{n \to \infty} \frac{1}{n^2} \left[ e^{1/n} + 2e^{2/n} + 3e^{3/n} + \dots + 2n e^{2n/n} \right] =\)

If \(\int \frac{dx}{(x^2+9)\sqrt{x^2+16}} = \frac{1}{3\sqrt{7}} \tan^{-1} \left( K \frac{x}{\sqrt{16+x^2}} \right) + c\), then \(K =\)