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integers.tex
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\documentclass{article}
\usepackage{amsmath,amssymb,gensymb}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{tfrupee}
\newcommand{\myvec}[1]{\ensuremath{\begin{pmatrix}#1\end{pmatrix}}}
\let\vec\mathbf
\begin{document}
\begin{enumerate}
\item Find the sum of the order and the degree of the differential equation
\begin{align}
\myvec{x + \frac{dy}{dx}}^2 = \myvec{\frac{dy}{dx}}^2 +1
\end{align}
\item If $\frac{d}{dx} [f(x)]$ = $\frac{\sec^4x}{\csc^4x}$ and $F\myvec{\frac{\pi}{4}} = \frac{\pi}{4}$, then find $F(x)$.
\item Find : $\int \frac {\log x-3}{(\log x)^4} dx$
\item Find : $\int \frac {dx}{\sqrt{x}+\sqrt[3]{x}}$
\item Evaluate : $\int_0^\frac{\pi}{2} \frac{\cos x}{(1+\sin x)(4+\sin x)} dx$
\item Evaluate : $\int_0^\pi \frac{x}{1+\sin x} dx$
\item Using integration, find the area of the region enclosed by the curve $y = x^2$, the x-axis and the ordinates $x=-2$ and $x=1$
\item Using integration, find the area of the region enclosed by the line $y = \sqrt{3}x$, semi-circle $y = \sqrt{4-x^2}$ and $x$-axis in first quadrant.
\item Find the product of the order and the degree of the differential equation $[\frac{d}{dx}(xy^2)].\frac{dy}{dx}+y=0$
\item Find : $\int \frac{\sqrt{\cot x}}{\sin x \cos x}dx$
\item Find : $\int \frac{1}{x(x^2+4)}dx$
\item Evaluate : $\int_0^1 \tan^{-1}x dx$
\item Find : $\int \frac{2x}{x^2+3x+2}dx$
\item Solve the following differential equation : $(1+e^{\frac{y}{x}}) dy + e^{\frac{y}{x}}\myvec{1-\frac{y}{x}} dx = 0$
\item Evaluate : $\int_0^1 x(1-x)^n dx$
\item Using integration, find the area of the smaller region enclosed by the curve $4x^2+4y^2=9$ and the line $2x+2y=3$
\item If the area of the region bounded by the curve $y^2=4ax$ and the line $x=4a$ is $\frac{256}{3}$ sq. units, then using integration, find the value of $a$, where $a>0$.
\item Find the general solution of the differential equation : $\frac{dy}{dx}=\frac{3e^{2x}+3e^{4x}}{e^x+e^{-x}}$
\item Find : $\int \frac{dx}{x^2-6x+13}$
\item Find the particular solution of the differential equation $x \frac{dy}{dx}-y=x^2.e^x$,given $y(1)=0$
\item Find the general solution of the differential equation
\begin{align}
x \frac{dy}{dx}=y(\log y-\log x +1)
\end{align}
\item Evaluate : $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\sin |x| + \cos |x|) dx$
\item Find : $\int \frac{x^2}{(x^2+1)(3x^2+4)} dx$
\item Evaluate : $\int_{-2}^1 \sqrt{5-4x-x^2}dx$
\item Find the area of the region enclosed by the curves $y^2=x, x=\frac{1}{4},y=0$ and $x=1$, using integration.
\item Evaluate :
\begin{align}
\int_0^1 x^2e^x dx
\end{align}
\item Find the general solution of the differential equation
\begin{align}
\sec^2 x . \tan y dx + \sec^2 y . \tan x dy=0
\end{align}
\item If the area of the region bounded by the line $y=mx$ and the curve $x^2=y$ is $\frac{32}{3}$ sq. units, then find the positive value of m, using integration.
\item Find :
\begin{align}
\int \frac{1}{e^x +1}dx
\end{align}
\item Evaluate :
\begin{align}
\int_1^4 \lbrace|x|+|3-x|\rbrace dx
\end{align}
\item Evaluate :
\begin{align}
\int_{-3}^3 \frac{x^4}{1+e^x}dx
\end{align}
\item Find the particular solution of the differential equation
$x\frac{dy}{dx}+y+\frac{1}{1+x^2}=0$, given that $y(1)=0$
\item Find the general solution of the differential equation
\begin{align}
x(y^3+x^3)dy=(2y^4+5x^3y)dx
\end{align}
\item Find : $\int \frac{dx}{\sqrt{4x-x^2}}$
\item Find the general solution of the following differential equation :
\begin{align}
\frac{dy}{dx}=e^{x-y}+x^2 e^{-y}
\end{align}
\item Let $X$ be a random variale which assumes values $x_1,x_2,x_3,x_4$ such that $2P(X=x_1)=3P(X=x_2)=P(X=x_3)=5P(X=x_4)$. Find the probability distribution of X.
\item Find :
\begin{align}
\int e^x . \sin 2x dx
\end{align}
\item Find :
\begin{align}
\int \frac{2x}{(x^2 +1)(x^2 +2)} dx
\end{align}
\item Evaluate : $\int_1^3 \frac{\sqrt{x}}{\sqrt{x} + \sqrt{4-x}}$
\item Solve the following differential equation:
\begin{align}
(y-\sin^2 x)dx+\tan x dy=0
\end{align}
\item Find the general solution of the differential equation:
\begin{align}
(x^3+y^3)dy=x^2y dx
\end{align}
\item Find : $\int \frac{1}{\sqrt{12+4x-x^2}}dx$
\item Find : $\int \frac{xe^x}{(x+4)^5}dx$
\item Find the general solution of the following differential equation :
\begin{align}
(4+y^2)(3+\log x)dx+x dy=0
\end{align}
\item Evaluate : $\int_0^\frac{\pi}{3} |\cos 3x| dx$
\item Find the general solution of the following differential equation :
\begin{align}
2xe^{\frac{y}{x}}dy + (x-2y e^{\frac{y}{x}})dx=0
\end{align}
\item Find the particular solution of the differentila equation $(2x^2+y).\frac{dx}{dy}=x$; given that $y=2$ when $x=1$
\item Find : $\int \frac{x^2+x+1}{(x+1)(x^2+4)}dx$
\item Find the area bounded by the ellipse $x^2+4y^2=16$ and the ordinates $x=0$ and $x=2$, using integration.
\item Find the area of the region $\lbrace (x,y) : x^2\leq y \leq x\rbrace$, using integration.
\item $\int_0^\frac{\pi}{2} \frac{1}{1+\sqrt{\cot x}} dx$ is equal to
\begin{enumerate}
\item $\frac{\pi}{3}$
\item $\frac{\pi}{6}$
\item $\frac{\pi}{4}$
\item $\frac{\pi}{2}$
\end{enumerate}
\item Find :
\begin{align}
\int \frac{(x+2)(x+2 \log x)^3}{x}dx
\end{align}
\item Evaluate :
\begin{align}
\int_0^\frac{\pi}{2} \log(\tan x)dx
\end{align}
\item Evaluate :
\begin{align}
\int_{-1}^2 |x| dx
\end{align}
\item Find :
\begin{align}
\int x^2 \log x . dx
\end{align}
\item Find the general solution of the following differential equation :
\begin{align}
\frac{dy}{dx}=(1+x)(1+y)
\end{align}
\item Find the integrating factor for the following differential equation :
\begin{align}
\frac{dy}{dx}+y\cot x=2x+x^2\cot x (x\neq0)
\end{align}
\item Find :
\begin{align}
\int \frac{x}{(x-1)^2(x+2)}dx
\end{align}
\item Solve the following differential equations :
\begin{align}
x\cos\myvec{\frac{y}{x}} \frac{dy}{dx} = y \cos\myvec{\frac{y}{x}}+x
\end{align}
\item If $\int \frac{\cos8x + 1}{\tan2x - \cot2x}dx = \lambda \cos8x + c$, then the value of $\lambda$ is
\begin{enumerate}
\item $\frac{1}{16}$
\item $\frac{1}{8}$
\item $\frac{-1}{16}$
\item $\frac{-1}{8}$
\end{enumerate}
\item $\int_0^1 \tan(\sin^{-1}x)dx$ equals
\begin{enumerate}
\item $2$
\item $0$
\item $-1$
\item $1$
\end{enumerate}
\item The integrating factor of the differential equation $x\frac{dy}{dx}-y=\log x$ is?
\item Find the solution of the differential equation $\log \frac{dy}{dx}=ax+by$.
\item Solve the following homogeneous differential equation :
\begin{align}
x\frac{dy}{dx}=x+y
\end{align}
\item Evaluate $\int_1^3 (x^2 + 1 + e^x)dx$ as the limit of sums.
\item If the area between the curves $x=y^2$ and $x=4$ is divided into two equal parts by the line $x=a$, then find the value of $a$ using integration.
\item Find :
\begin{align}
\int \frac{x}{(x-1)^2(x+2)}dx
\end{align}
\item Evaluate :
\begin{align}
\int_0^1 \frac{xe^x}{(x+1)^2}dx
\end{align}
\item Solve the following differential equation :
\begin{align}
\frac{dy}{dx} = e^{x+y} + x^2 e^y
\end{align}
\item The supply function of a commodity is $100p=(x+20)^2$. Find the Producer's Surplus (PS), when the market price is \rupee~25
\item Find :
\begin{align}
\int \frac{2x^2 +1}{x^2 - 3x +2}dx
\end{align}
\item In a certain culture of bacteria, the rate of increase of bacteria is proportional to the number present. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours. determine the numer of bacteria present in the beginning.
\end{enumerate}
\end{document}