List of practice Questions

Using integration, find the area of the region enclosed by the curve \( y = |x - 6| \), the x-axis, and between \( x = 4 \) and \( x = 8 \).

\( \int_{-1}^{1} (1 - |x|) \, dx \) is equal to :

Evaluate : \( \int_{-\pi/6}^{\pi/2} (\sin |x| + \cos |x|) dx \)

\( \int \frac{dx}{\sec x + \tan x} \) is equal to

If \( I_1 = \int_{-\pi/4}^{\pi/4} \frac{dx}{1 + \cos 2x} \) and \( I_2 = \int_{-1/2}^{1/2} |x| dx \), then show that \( I_1 - 4I_2 = 0 \).

The area bounded by the curve \( y = x|x| \), x-axis and the ordinates \( x = -1 \) and \( x = 1 \) is given by

If \( \int \frac{3ax}{b^2 + c^2x^2} dx = A \log |b^2 + c^2x^2| + K \), then the value of \( A \) is

The value of \( \int_{-1}^{1} \frac{x^3}{x^2 + 2|x| + 1} dx \) is

The correct integral of \(\displaystyle\int \frac{\sin x}{\sqrt{1+\cos x}}\,dx\) (assuming integral constant as \(c\)) is ______.

\(\int \frac{1}{2\cot x-3\tan x}dx =\)

Which of the following functions has the highest area under the curve between \(x=0\) and \(x=10\)?

Evaluate \[ \int \frac{1}{(x^4+1)^{5/4}}\,dx. \]

The integral
\[ \frac{1}{\pi}\int_0^{\infty}\frac{x^{2026}}{(1+x^{2026})(1+x^2)}\,dx \] evaluates to (round off to two decimal places).

Evaluate \[ \int \frac{\sin^2x\tan x} {\cos^6x+\sin^4x\cos^2x+\cos^4x\sin^2x+\sin^6x}\,dx. \]

Evaluate \[ \int \frac{x^7-1} {x^8\sqrt{\,1-2x^7+7x^{14}\,}} \,dx. \]

If \[ f(x)=\frac{x^3}{1-3x+3x^2}, \] then \[ \lim_{n\to\infty} \frac{2}{n} \sum_{r=1}^{n} f\!\left(\frac{r}{n}\right) = \]

Let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be an odd function and \[ \int_{-1}^{1}x^3f''(x)\,dx=\frac85. \] If \[ g(x)=xf(x), \] \[ g'(1)=\frac83 \] and \[ \int_0^1g(x)\,dx=\frac{2}{15}, \] then \(f(1)=\)

The area of the region bounded by the curves \[ y=x^2+1,\qquad x=y^2+1, \] \(X\)-axis, \(Y\)-axis and \(x=2\) (in sq. units) is

If \[ \int \frac{e^{\cos x}\sin x}{e^{\cos x}+e^{-\cos x}}\,dx = -\frac12\left[f(x)+\log\!\left(e^{\cos x}+e^{-\cos x}\right)\right]+c \] and \[ f\!\left(\frac{\pi}{3}\right)=\frac12, \] then the number of points at which \(f(x)\) attains the maximum value in \[ [-2\pi,2\pi] \] is

If \[ \int \frac{\cos\left(\frac{7x}{2}\right)}{\cos\left(\frac{x}{2}\right)}\,dx = a\sin x+b\sin2x+c\sin^3x-dx+k, \] then \(a+b+c+d=\)

Let \(f(x)\) be a continuous function defined in \([0,2] \to \mathbb{R}\) and satisfying the equation
\[ \int_0^2 f(x)[x - f(x)]\,dx = \frac{2}{3} \]
The value of \(f(1)\) is

The value of \[ \int_{0}^{\infty}\frac{x^{2}(1-x^{2})}{(1+x)^{24}}\,dx \] is

The value of \[ \int_{0}^{\infty}\frac{x^{2}(1-x^{2})}{(1+x)^{24}}\,dx \] is

If \[ \int \frac{dx}{2\sin^2x+\sin2x} = \frac12\log|f(x)|+C \] and \[ f\!\left(\frac{\pi}{4}\right)=\frac12, \] then \(f(x)\) is

If \[ \int \frac{2\cos x+3\sin x} {3\cos x+4\sin x}\,dx = Ax+B\log|3\cos x+4\sin x|+C, \] then \[ A\cdot B= \]