\begin{align} \int \frac{x^3 + 5x^2 - 4}{x^2} . dx\end{align}
\begin{align} =\int \left(x + 5 - 4x^{-2}\right) . dx\end{align}
\begin{align} =\int x .dx + 5 \int 1.dx- 4 \int x^{-2} .dx\end{align}
\begin{align} =\frac {x^2}{2} + 5x + 4 \left(\frac{x^{-1}}{-1}\right) + C\end{align}
\begin{align} =\frac {x^2}{2} + 5x + \frac{4}{x} + C\end{align}
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
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
(A) 10π (B) 12π (C) 8π (D) 11π
If f(x) = , show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Determine order and degree(if defined) of differential equation y' + 5y = 0
Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f–1 of f given by , where R+ is the set of all non-negative real numbers.
Show that the Signum Function f : R → R, given by
is neither one-one nor onto.