First Fundamental Theorem of Calculus

Q. ${\int }_0^{\frac{\pi }{6}}(2+3x^2)cos3xdx=$

1. $\frac{1}{36}(\pi +16)$
2. $\frac{1}{36}(\pi -16)$
3. $\frac{1}{36}({\pi }^2-16)$
4. $\frac{1}{36}({\pi }^2+16)$

Q. Let f(x) be a function satisfying f ’(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = $x^2$. Then, the value of the integral ${\int }_0^{1}f\left(x\right)g\left(x\right)dx,$ is

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

Q. ${\int }_0^1{e}^{2Inx}dx=$

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

Q. If g(x) = ${\int }_0^x$ cos 4 tdt, then g(x +$\pi$ ) equals

1. $g\left(x\right)$
2. $g\left(x\right).g\left(\pi \right)$
3. $\frac{g\left(x\right)}{g\left(\pi \right)}$
4. None of these

Q. ${\int }_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}}\frac{dx}{\sqrt{4-9x^2}}dxis$

1. $\frac{\pi }{6}$
2. $\frac{\pi }{12}$
3. $\frac{\pi }{24}$
4. $\frac{\pi }{36}$

Q. ${\int }_0^{2\pi }\sqrt{1+sin\frac{x}{2}}dx=$

1. \N
2. 2
3. 8
4. 4

Q. The value of the definite integral $\underset{0}{\overset{\pi }{\int }}\frac{\pi \tan x}{\sec x+\tan x}dx$ is equal to

1. $\pi (1-\pi )$
2. $\pi (\pi -2)$
3. $\pi (2-\pi )$
4. $\pi (\pi -1)$

Q. If f(y) =$e^y$, g(y) = y ; y > 0 and F(t) = ${\int }_0^{t}f(t-y)$ g(y) dy, then

1. $F\left(t\right)=e^t-(1+t)$
2. $F\left(t\right)=t{e}^t$
3. $F\left(t\right)=t{e}^{-t}$
4. $F\left(t\right)=1-e^{-t}(1+t)$

Q. Fundamental period of the function $f\left(x\right)=\frac{(1+\sin x)(1+\sec x)}{(1+\cos x)(1+\text{cosec}x)},x\in R-\left\{\right(2n+1)\pi ,\frac{(4m-1)\pi }{2},n,m\in Z\}$ is

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

Q. ${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{e}^x(logsinx+cotx)dx=$

1. $e^{\frac{\pi }{4}}log2$
2. $-e^{\frac{\pi }{4}}log2$
3. $\frac{1}{2}{e}^{\frac{\pi }{4}}log2$
4. $-\frac{1}{2}{e}^{\frac{\pi }{4}}log2$

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

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

Q. $If{\int }_0^1{e}^{x^2}(x-\alpha )dx=0,then$

1. $1<\alpha <2$
2. $\alpha <0$
3. $0<\alpha <1$
4. $\alpha =0$

Q. ${\int }_1^e\frac{e^x}{x}(1+xlogx)dx=$

1. $e^e$
2. $e^e-e$
3. $e^e+e$
4. $Noneofthese$

Q. If $l(m,n)={\int }_0^1{t}^m(1+t{)}^{n}dt$, then the expression for $l(m,n)$ in terms of $l(m+1,n-1)$ is

1. $\frac{2^n}{m+1}-\frac{n}{m+1}l(m+1,n-1)$
2. $\frac{n}{m+1}l(m+1,n-1)$
3. $\frac{2^n}{m+1}+\frac{n}{m+1}l(m+1,n-1)$
4. $\frac{m}{m+1}l(m+1,n-1)$

Q. The maximum value of the function
$f\left(x\right)=\underset{0}{\overset{1}{\int }}t\sin (x+\pi t)dt,x\in R$ is

1. $\frac{1}{\pi }\sqrt{{\pi }^2+4}$
2. $\frac{1}{{\pi }^2}\sqrt{{\pi }^2+4}$
3. $\sqrt{{\pi }^2+4}$
4. $\frac{1}{2{\pi }^2}\sqrt{{\pi }^2+4}$

Q. ${\int }_0^{1}co{s}^{-1}xdx=$

1. \N
2. 1
3. 2
4. None of these

Q. ${\int }_0^{\infty }(a^{-x}-b^{-x})dx=$

1. $\frac{1}{loga}-\frac{1}{logb}$
2. $loga-logb$
3. $loga+logb$
4. $\frac{1}{loga}+\frac{1}{logb}$

Q. ${\int }_0^{\frac{\pi }{8}}\frac{se{c}^{2}2x}{2}dx=$

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

Q. ${\int }_a^b\frac{logx}{x}dx=$

1. $log\left(\frac{logb}{loga}\right)$
2. $log\left(ab\right)log\left(\frac{b}{a}\right)$
3. $\frac{1}{2}log\left(ab\right)log\left(\frac{b}{a}\right)$
4. $\frac{1}{2}log\left(ab\right)log\left(\frac{a}{b}\right)$

Q. The value of integral ${\int }_1^{e^6}\left[\frac{logx}{3}\right]dx$, where [.] denotes the greatest integer function, is

1. $0$
2. $e^6-e^3$
3. $e^6+e^3$
4. $e^3-e^6$

Q. ${\int }_{-1}^1{\{{\left(\frac{x+2}{x-2}\right)}^2+{\left(\frac{x-2}{x+2}\right)}^2-2\}}^{\frac{1}{2}}dx=$

1. $8log\frac{4}{3}$
2. $8log\frac{3}{4}$
3. $4log\frac{4}{3}$
4. $4log\frac{3}{4}$

Q. If $f\left(x\right)$ is a function satisfying $f^{\prime }\left(x\right)=f\left(x\right)$ with $f\left(0\right)=1$ and $g\left(x\right)$ be another function such that $f\left(x\right)+g\left(x\right)=x^2$, then the value of $\underset{0}{\overset{1}{\int }}f\left(x\right)g\left(x\right)dx$ is

1. $\frac{1}{4}(e-e^2-1)$
2. $\frac{1}{4}(e-2)$
3. $\frac{1}{2}(e^2-e-3)$
4. $\frac{1}{2}(2e-e^2-3)$

Q. Fundamental period of $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$ is

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

Q. ${\int }_0^{\frac{\pi }{4}}ta{n}^{2}xdx=$

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

Q.

${\int }_1^{a}x.a^{-\left[lo{g}_{a}x\right]}dx=\frac{e-1}{2},\text{where}a>1\text{and}[.]$ denotes the greatest integer function, then the value of $a^2$is

1. $e-1$

2. $e$

3. $e+1$

4. $e^2-1$

Q. ${\int }_0^{\frac{\pi }{8}}\frac{se{c}^{2}2x}{2}dx=$

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

Q. ${\int }_1^2{e}^x(\frac{1}{x}-\frac{1}{x^2})dx=$

1. $\frac{e^2}{2}+e$
2. $e-\frac{e^2}{2}$
3. $\frac{e^2}{2}-e$
4. $Noneofthese$