Improper Integrals

Q. Suppose f, g are continuous and differentiable on [0, b], are non-negative on [0, b] and f is non constant with f(0) = 0, then the minimum value of ${\int }_0^{a}g\left(x\right)f^{\prime }\left(x\right)dx+{\int }_0^b{g}^{\prime }\left(x\right)f\left(x\right)dxonaϵ(0,b]$ is

1. f(a).g(a)
2. f(b).g(b)
3. f(a).g(b)
4. f(b).g(a)

Q. The value of the integral ${\int }_0^{\infty }\frac{x\mathrm{dx}}{\left(1+x\right)\left(1+x^2\right)}$ is equal to

1. $\pi$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $2\pi$

Q. $\mathrm{If}{\int }_0^{\infty }\frac{\sin x}{x}\mathrm{dx}=\frac{\pi }{2},\mathrm{then}{\int }_0^{\infty }\frac{{\sin }^{3}x}{x}\mathrm{dx}$
is equal to

1. $\frac{\pi }{2}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{6}$
4. $\frac{3\pi }{2}$

Q. If $f\left(x\right)=\sum _{k=1}^x{\tan }^{-1}\left(\frac{2k}{2+k^2+k^4}\right),$ then the value of $6f^{\prime }\left(0\right)$ is

Q. the value of tan inverse a + tan inverse 1/a , where a<0

Q.

The differential coefficient of the function f(x) = a^{sin x}, where a is positive constant is:

1. cos x. log sin x

2. a^{sin x} log sin x

3. a^{sin x}. cos x. log sin x

4. a^{sin x}. cos x. log a

Q. Number of Partial Fractions of $\frac{2x+3}{x^2(x^2-1{)}^3}$

1. 5
2. 7
3. 8
4. 6

Q. The value of the integral ${\int }_0^{\infty }\frac{x\mathrm{dx}}{\left(1+x\right)\left(1+x^2\right)}$ is equal to

1. $\pi$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $2\pi$

Q.

${\int }_{\frac{1}{-\sqrt{2}}}^{\frac{1}{\sqrt{2}}}\left[{\left(\frac{1-x}{1+x}\right)}^2+{\left(\frac{1+x}{1-x}\right)}^2-2\right]dx$

Q. Obtain reduction formula : $\int {\tan }^{n}xdx$
for integer $n\ge 2$ and evaluate : $\int {\tan }^{6}xdx$

Q. The range of the functions $f\left(x\right)={\int }_1^x|t|dt,xϵ[\frac{-1}{2},\frac{1}{2}]$ is

1. $[\frac{3}{8},\frac{5}{8}]$
2. $[\frac{-5}{8},\frac{3}{8}]$
3. $[\frac{-3}{8},\frac{5}{8}]$
4. $[\frac{-5}{8},\frac{-3}{8}]$

Q. The function $f\left(x\right)=\{\begin{array}{cc}|x-3|& ,x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4}& ,x<1\end{array}$ is

1. not continuous at $x=1$
2. not differentiable at $x=1$
3. continuous and differentiable at $x=1$
4. continuous at $x=1$ but not differentiable at $x=1$

Q. $f\left(x\right)=\{\begin{array}{c}{x}^{2}for0\le x\le 1\\ \sqrt{x}for1\le x\le 2\end{array}then{\int }_0^{2}f\left(x\right)xdx=$

1. $\left(\frac{1}{3}\right)(4\sqrt{2}-1)$
2. $\left(\frac{1}{3}\right)(2\sqrt{2}-1)$
3. $\left(\frac{1}{2}\right)(3\sqrt{2}-1)$
4. $\left(\frac{1}{3}\right)(4\sqrt{2}+1)$

Q. The solution of the inequality $({\tan }^{-1}x{)}^2-3{\tan }^{-1}x+2\ge 0$ is -

1. $(-\infty ,\tan 1]$
2. $(-\infty ,\tan 1]\cup [\tan 2,\infty )$
3. $(-\infty ,-\tan 1]\cup [\tan 2,\infty )$
4. $[\tan 2,\infty )$

Q.

Let a, b, c be non-zero real numbers such that
${\int }_0^1(1+co{s}^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+co{s}^{8}x)(a{x}^2+bx+c)dx$
Then the quadratic equation $a{x}^2+bx+c=0$ has [IIT 1981; CEE 1993]

1. No root in (0, 2)

2. At least one root in (0, 2)

3. None of these

4. A double root in (0, 2)

Q. If ${\tan }^{-1}a+{\tan }^{-1}b+{\tan }^{-1}c=x$, then prove that $a+b+c=abc$.

Q. The sum of the series ${\tan }^{-1}\frac{1}{1+1+1^2}+{\tan }^{-1}\frac{1}{1+2+2^2}+{\tan }^{-1}\left(\frac{1}{1+3+3^2}\right)+......\infty$

1. $\frac{\pi }{3}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. None of these

Q. If $r^2=x^2+y^2+z^2$, then prove that
${\tan }^{-1}\left(\frac{yz}{rx}\right)+{\tan }^{-1}\left(\frac{zx}{ry}\right)+{\tan }^{-1}\left(\frac{xy}{rz}\right)=\frac{\pi }{2}$

Q.

Let f and g are two real valued differentiable functions satisfying.
$f\left(x\right)=\underset{Lt}{\alpha \to 0}\frac{1}{{\alpha }^4}{\int }_0^{\alpha }\frac{(e^{x+t}-e^x)\left(l{n}^2\right(t+1\left)\right)}{2t^2+3}dtand{\int }_0^{x}g\left(t\right)dt=3x+{\int }_x^{0}co{s}^{2}tg\left(t\right)dt$
Range of g(x) is

1. $[\frac{3}{2},\infty )$

2. $[\frac{3}{2},3]$

3. $[-3,3]$

4. $[-\frac{3}{2},\frac{3}{2}]$

Q. If$x^{2}y=y-2x$ and $x^2<1$. then $a\times [x+\frac{1}{3}{x}^3+\frac{1}{5}{x}^5+.....]=y+\frac{1}{3}{y}^3+\frac{1}{5}{y}^5+......$ find $a$

Q. Resolve $\frac{(2+cosx)}{(1-cosx)(3+cosx)(1+cosx{)}^2}$ into partial fractions.

1. $\frac{3}{8(1-cosx)}-\frac{1}{8(3+cosx)}+\frac{1}{4(1+cosx)}+\frac{4}{4(1+cosx{)}^2}$
2. $\frac{3}{16(1-cosx)}-\frac{1}{16(3+cosx)}+\frac{1}{8(1+cosx)}+\frac{4}{4(1+cosx{)}^2}$
3. $\frac{3}{16(1-cosx)}-\frac{1}{16(3+cosx)}+\frac{1}{4(1+cosx)}+\frac{4}{8(1+cosx{)}^2}$
4. $\frac{3}{16(1-cosx)}-\frac{1}{16(3+cosx)}+\frac{1}{4(1+cosx)}+\frac{4}{4(1+cosx{)}^2}$

Q. The value of ${\int }_0^1{\tan }^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is

1. $1$
2. $0$
3. $-1$
4. $\frac{\pi }{4}$

Q. Differentiate ${\tan }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ with respect to ${\sin }^{-1}\left(2x\sqrt{1-x^2}\right),\mathrm{if}-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$

Q. If $f$ is a continuous function on $[0,2],$ differentiable in $(0,2)$ such that $f\left(2\right)=0,$ then which of the following options is(are) correct for atleast one value of $c\in (0,2)$

1. $c^2{f}^{\prime }\left(c\right)+cf\left(c\right)=0$
2. $c^2{f}^{\prime }\left(c\right)+2cf\left(c\right)=0$
3. $c^3{f}^{\prime }\left(c\right)+c^{2}f\left(c\right)=0$
4. $c^3{f}^{\prime }\left(c\right)+3c^{2}f\left(c\right)=0$

Q. $\int \frac{dx}{(1+x^2)\sqrt{p^2+q^2(ta{n}^{-1}x{)}^2}}=$

1. $\frac{1}{q}log[qta{n}^{-1}x+\sqrt{p^2+q^2(ta{n}^{-1}x{)}^2}]+c$
2. $\frac{2}{3q}(p^2+q^{2}ta{n}^{-1}x{)}^{3/2}+c$
3. $log[qta{n}^{-1}x+\sqrt{p^2+q^2(ta{n}^{-1}x{)}^2}]+c$
4. None of these

Q. lf $\int \frac{dx}{(x^2+1)(x^2+4)}=$ $A{\tan }^{-1}+B{\tan }^{-1}\left(\frac{x}{2}\right)+C$, then $B=$

1. $6$
2. $1/6$
3. $-1/6$
4. $-6$

Q. Differentiate ${\sin }^{-1}\sqrt{1-x^2}$ with respect to ${\cot }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right),\mathrm{if}0<x<1$

Q. Resolve $\frac{2x^3-3x^2-8x-26}{2x^2-5x-12}$ into partial fractions.

1. $x+1+\frac{3}{2x+3}+\frac{2}{x-4}$
2. $x-1+\frac{5}{2x+3}+\frac{2}{x-4}$
3. $x+1+\frac{5}{2x+3}+\frac{2}{x-4}$
4. $x+\frac{5}{2x+3}+\frac{2}{x-4}$

Q. If $\alpha$ and $\beta$ are the roots of the equation ${ax}^2+bx+c=0$ ($a\ne 0,a.b,c$ being different), then $(1+\alpha +{\alpha }^2)(1+\beta +{\beta }^2)$ is equal to

1. Zero
2. Positive
3. Negative
4. None of these

Q. ${\int }_{-1}^3(Ta{n}^{-1}\frac{x}{x^2+1}+Ta{n}^{-1}\frac{x^2+1}{x})dx=$

1. $\pi$
2. $2\pi$
3. $4\pi$
4. $none$