Question

Q1.
Integration of x2 + 2x with res pect to x will
be -
2
+3
(A)
+ x² + c
(B) 2x + 2
3
x
(C)
x² - c
(D) 2x + 2 + c​

Answer 1

Step-by-step explanation:

Solution :

$\begin{gathered}:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx} \\ \\ \end{gathered} </p><p>$

Now by using the sum of integration rule, we get

$\begin{gathered}\underline{\sf{\bigstar\:Sum\:rule\:of\: Integration :-}} \\ \\ :\implies \displaystyle\int \sf[f(x) + g(x)]dx = \displaystyle\int \sf f(x)dx + \displaystyle\int \sf g(x)dx \\ \\ \end{gathered} </p><p></p><p>$

$</p><p>\begin{gathered}:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \displaystyle\int \sf x^{2}dx + \displaystyle\int \sf2xdx} \\ \\ \end{gathered} </p><p>$

Now,

Integration of x² :

$\begin{gathered}:\implies \displaystyle\int \sf x^{2}dx \\ \\ \end{gathered} </p><p>$

Now by using the power of integration rule, we get :

$\begin{gathered}\underline{\sf{\bigstar\: Power\:rule\:of\: Integration :-}} \\ \\ :\implies \displaystyle\int \sf x^{n}dx = \dfrac{x^{(n + 1)}}{n + 1} + C \\ \\ \end{gathered} </p><p>$

$\begin{gathered}:\implies \displaystyle\int \sf x^{2}dx = \dfrac{x^{(2 + 1)}}{2 + 1} + C \\ \\ \end{gathered} </strong></p><p><strong>[tex]\begin{gathered}:\implies \displaystyle\int \sf x^{2}dx = \dfrac{x^{(2 + 1)}}{2 + 1} + C \\ \\ \end{gathered}$

$\begin{gathered}:\implies \displaystyle\int \sf x^{2}dx = \dfrac{x^{3}}{3} + C \\ \\ \end{gathered} </p><p>$

$\begin{gathered}\underline{\boxed{ \displaystyle\int \sf x^{2}dx = \dfrac{x^{3}}{3} + C}} \\ \\ \end{gathered} </p><p>$

Integration of 2x :

$\begin{gathered}:\implies \displaystyle\int \sf 2xdx \\ \\ \end{gathered} </p><p></p><p>$

$</p><p>\begin{gathered}:\implies \displaystyle\int \sf 2xdx = 2\displaystyle\int \sf xdx \\ \\ \end{gathered} </p><p></p><p> </p><p>$

Now by using the power of integration rule, we get

$\begin{gathered}\underline{\sf{\bigstar\: Power\:rule\:of\: Integration :-}} \\ \\ :\implies \displaystyle\int \sf x^{n}dx = \dfrac{x^{(n + 1)}}{n + 1} + C \\ \\ \end{gathered} </p><p>$

$</p><p>\begin{gathered}:\implies 2\displaystyle\int \sf x^{2}dx = 2 \times \left(\dfrac{x^{(1 + 1)}}{1 + 1}\right) + C \\ \\ \end{gathered} </p><p>$

$\begin{gathered}:\implies 2\displaystyle\int \sf x^{2}dx = 2 \times \dfrac{x^{2}}{2} + C \\ \\ \end{gathered} </p><p>$

$\begin{gathered}:\implies \displaystyle\int \sf x^{2}dx = x^{2} + C \\ \\ \end{gathered} </p><p>$

$\begin{gathered}\underline{\boxed{ \displaystyle\int \sf 2xdx = x^{2} + C}} \\ \\ \end{gathered} </p><p>$

Now by substituting the integration of x² and 2x in the equation, we get :

$</p><p>\begin{gathered}:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \displaystyle\int \sf x^{2}dx + \displaystyle\int \sf2xdx} \\ \\ \end{gathered} </p><p>$

$\begin{gathered}:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \dfrac{x^{3}}{3} + x^{2} + C } \\ \\ \end{gathered} </p><p>$

$\begin{gathered}\underline{\boxed{ \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \dfrac{x^{3}}{3} + x^{2} + C}}} \\ \\ \end{gathered} </p><p>$

Hence the integration of (x² + 2x) w.r.t x is (x³/3 + x² + C).