JEE Main Mathematics — Calculus previous year questions with solutions.
Let $f$ be a real valued function, defined on$R-{-1,1}$ and given by $f(x)=3{\mathrm{log}}_{e}|\frac{x-1}{x+1}|-\frac{2}{x-1}$. Then in which of the following intervals, function $f(x)$ is increasing?
The minimum value of $\alpha$ for which the equation $\frac{4}{\mathrm{sin}x}+\frac{1}{1-\mathrm{sin}x}=\alpha$ has at least one solution in $(0,\frac{\pi }{2})$ is______.
If the value of $\underset{x\rightarrow 0}{\mathrm{lim}}{(2-\mathrm{cos}x\sqrt{\mathrm{cos}2x})}^{(\frac{x+2}{{x}^{2}})}$ is equal to ${e}^{a}$, then $a$ is equal to_____.
Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f(x)=2{x}^{3}-3{x}^{2}-12x$. If $A$ is the total area of the region bounded by $y=f(x),$ the $x$-axis and the lines $x=a$ and $x=b,$ then $4A$ is equal to ______.
The number of points, at which the function $f(x)=|2x+1|-3|x+2|+|{x}^{2}+x-2|,x\in R$ is not differentiable, is
The integral $\int \frac{1}{\sqrt[4]{(x-1{)}^{3}(x+2{)}^{5}}}dx$ is equal to : (where $C$ is a constant of integration)
If $\int \frac{\mathrm{sin}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}dx=\alpha {\mathrm{log}}_{e}|1+\mathrm{tan}x|+\beta {\mathrm{log}}_{e}|1-\mathrm{tan}x+{\mathrm{tan}}^{2}x|+\gamma {\mathrm{tan}}^{-1}(\frac{2\mathrm{tan}x-1}{\sqrt{3}})+C,$ when $C$ is constant of integration, then the value of $18(\alpha +\beta +{\gamma }^{2})$ is
The integral $\int \frac{(2x-1)\mathrm{cos}\sqrt{(2x-1{)}^{2}+5}}{\sqrt{4{x}^{2}-4x+6}}dx$ is equal to (where $c$ is a constant of integration)
The integral $\int \frac{{e}^{3{\mathrm{log}}_{e}2x}+5{e}^{2{\mathrm{log}}_{e}2x}}{{e}^{4{\mathrm{log}}_{e}x}+5{e}^{3{\mathrm{log}}_{e}x}-7{e}^{2{\mathrm{log}}_{e}x}}dx,x>0$, is equal to (where $c$ is a constant of integration)
If $x\phi (x)={\int }_{5}^{x}(3{t}^{2}-2{\phi }^{'}(t))dt,x>-2,$ $\phi (0)=4,$ then $\phi (2)$ is
The value of the integral ${\int }_{0}^{1}\frac{\sqrt{x}dx}{(1+x)(1+3x)(3+x)}$ is:
Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1).$ If ${\int }_{0}^{x}\sqrt{1-{({f}^{'}(t))}^{2}}\mathrm{dt}={\int }_{0}^{x}f(t)\mathrm{dt},0\leq x\leq 1$ and $f(0)=0,$ then $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{1}{{x}^{2}}{\int }_{0}^{x}f(t)\mathrm{dt}:$
${\int }_{6}^{16}\frac{{\mathrm{log}}_{e}{x}^{2}}{{\mathrm{log}}_{e}{x}^{2}+{\mathrm{log}}_{e}({x}^{2}-44x+484)}dx$ is equal to
The value of the definite integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{dx}{(1+{e}^{x\mathrm{cos}x})({\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x)}$ is equal to :
If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral ${\int }_{-\pi /2}^{\pi /2}[[x]-\mathrm{sin}x]dx$ is equal to:
The value of the definite integral ${\int }_{\pi /24}^{5\pi /24}\frac{dx}{1+\sqrt[3]{\mathrm{tan}2x}}$ is
Let $a$ be a positive real number such that ${\int }_{0}^{a}{e}^{x-[x]}dx=10e-9$ where, $[x]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to:
Let $f:R\rightarrow R$ be defined as $f(x)={e}^{-x}\mathrm{sin}x.$ If $F:[0,1]\rightarrow R$ is a differentiable function such that $F(x)={\int }_{0}^{x}f(t)dt,$ then the value of ${\int }_{0}^{1}({F}^{'}(x)+f(x)){e}^{x}dx$ lies in the interval
If the normal to the curve $y(x)={\int }_{0}^{x}(2{t}^{2}-15t+10)dt$ at a point $(a,b)$ is parallel to the line $x+3y=-5,a>1$, then the value of $|a+6b|$ is equal to ________.
If ${I}_{m,n}={\int }_{0}^{1}{x}^{m-1}{(1-x)}^{n-1}dx$, for $m,n\geqslant 1$, and ${\int }_{0}^{1}\frac{{x}^{m-1}+{x}^{n-1}}{{(1+x)}^{m+n}}dx=\alpha {I}_{m,n},\alpha \in R$, then $\alpha$ equals ________.
For $x>0$, if $f(x)={\int }_{1}^{x}\frac{{\mathrm{log}}_{e}t}{(1+t)}dt$, then $f(e)+f(\frac{1}{e})$ is equal to
The value of the integral ${\int }_{0}^{\pi }|\mathrm{sin}2x|dx$ is ________.
The area of the region $S={(x,y):3{x}^{2}\leq 4y\leq 6x+24}$ is______.
If the line $y=mx$ bisects the area enclosed by the lines $x=0,y=0,x=\frac{3}{2}$ and the curve $y=1+4x-{x}^{2},$ then $12m$ is equal to .