\begin{align} \int \frac{x^3 + 3x + 4}{\sqrt{x}} . dx\end{align}
\begin{align} =\int \left(x^\frac{5}{2} + 3x^\frac{1}{2} + 4x^\frac{1}{2}\right) . dx\end{align}
\begin{align} =\frac{\left(x^{\displaystyle\frac72}\right)}{\displaystyle\frac72}+ \frac{3\left(x^{\displaystyle\frac32}\right)}{\displaystyle\frac32} + \frac{4\left(x^{\displaystyle\frac12}\right)}{\displaystyle\frac12} + C\end{align}
\begin{align} =\frac27\left(x^\frac72\right)+ 2\left(x^\frac32\right) + 8\left(x^\frac12\right) + C\end{align}
\begin{align} =\frac27\left(x^\frac72\right)+ 2\left(x^\frac32\right) + 8\left(\sqrt x\right) + C\end{align}
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}
Represent graphically a displacement of 40 km, 30° east of north.
If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Let f, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f . g)oh = (foh) . (goh)
Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.
(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,
)
\begin{align} y= \sqrt{1+x^2} : y^{'}=\frac{xy}{1+x^2}\end{align}