List of practice Questions

The value of the integral \(\int \frac{dx}{\cos 2x + \sin 2x + 2\sin^2 x}\) is:

If \(x > 0\), then \(\int \frac{1}{\sqrt{x^4 + 2x^3 + 2x^2}} dx =\)

The integral \( \int \frac{\cos^3 x}{(1+\sin x)^4} dx \) is equal to:

The integral \( \int \frac{\cos x - \sin x}{10 + \sin 2x} dx \) is equal to:

If $\int\frac{1}{x^{2}+4x+\alpha}dx=\frac{1}{2\sqrt{2}}\tan^{-1}\left(\frac{x+2}{2\sqrt{2}}\right)+c$, then $\int\frac{1}{x^{2}+4x-\alpha}dx=$

If $\int\frac{2\sin x+a\cos x}{b\sin x+4\cos x}dx=\frac{2}{5}x-\frac{1}{5}\log(b\sin x+4\cos x)+c$, then $a+b=$

If $\int(4\sin\theta+\cos\theta)\cot\theta\cos\theta d\theta=2\theta+\sin2\theta+\log(\sin\theta)+f(2\theta)+c$ and $f(0)=\frac{1}{2}$, then $f(x)=$

If $I_{n}=\int\sin^{n}x dx$, then $I_{6}-\frac{5}{6}I_{4}=$

Evaluate: \[ \int \sin^{-1}x\,dx-\int \cos^{-1}x\,dx. \]

If \[ \int\frac{dx}{\sqrt{4x^{2}+11x+6}} = \frac12\cosh^{-1}\!\left(\frac{f(x)}{5}\right)+C \] and \[ f(1)=19, \] then \(f(2)=\) ?

If \[ \int \log x\sqrt{\left(\frac{\log x}{x}\right)^2+\frac1{x^2}}\,dx = \frac{f(x)}{3}\sqrt{1+(\log x)^2}+C \] and \(f(1)=1\), then \(f(e)=\) ?

If \[ \int\frac1{1+\cos x}\,dx = \frac1{f\!\left(\frac x2\right)}+C_1, \] then \[ \int f(x)\,dx= \ ? \]

Let \[ f:[0,1]\rightarrow\mathbb R \] be a function defined by \[ f(x)+f(1-x)=1. \] Then \[ \int_0^1 f(x)\,dx= \ ? \]

If \([t]\) denotes the greatest integer function, then \[ \int_{-2}^{2} \left[ \frac{x^2+[x+1]} {1+x^2} \right]dx = \ ? \]

The area of the region bounded by the curve \[ y=|x-2|+|x-8|, \] the X-axis and the lines \(x=0\) and \(x=10\) 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