Question

27.
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

Solution :

$:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx}$

Now by using the sum of integration rule, we get

$\underline{\sf{\bigstar\:Sum\:rule\:of\: Integration :-}} \\ \\ :\implies \displaystyle\int \sf[f(x) + g(x)]dx = \displaystyle\int \sf f(x)dx + \displaystyle\int \sf g(x)dx$

$:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \displaystyle\int \sf x^{2}dx + \displaystyle\int \sf2xdx}$

Now,

• Integration of x² :

$:\implies \displaystyle\int \sf x^{2}dx$

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

$\underline{\sf{\bigstar\: Power\:rule\:of\: Integration :-}} \\ \\ :\implies \displaystyle\int \sf x^{n}dx = \dfrac{x^{(n + 1)}}{n + 1} + C$

$:\implies \displaystyle\int \sf x^{2}dx = \dfrac{x^{(2 + 1)}}{2 + 1} + C$

$:\implies \displaystyle\int \sf x^{2}dx = \dfrac{x^{3}}{3} + C$

$\underline{\boxed{ \displaystyle\int \sf x^{2}dx = \dfrac{x^{3}}{3} + C}}$

• Integration of 2x :

$:\implies \displaystyle\int \sf 2xdx$

$:\implies \displaystyle\int \sf 2xdx = 2\displaystyle\int \sf xdx$

Now by using the power of integration rule, we get

$\underline{\sf{\bigstar\: Power\:rule\:of\: Integration :-}} \\ \\ :\implies \displaystyle\int \sf x^{n}dx = \dfrac{x^{(n + 1)}}{n + 1} + C$

$:\implies 2\displaystyle\int \sf x^{2}dx = 2 \times \left(\dfrac{x^{(1 + 1)}}{1 + 1}\right) + C$

$:\implies 2\displaystyle\int \sf x^{2}dx = 2 \times \dfrac{x^{2}}{2} + C$

$:\implies \displaystyle\int \sf x^{2}dx = x^{2} + C$

$\underline{\boxed{ \displaystyle\int \sf 2xdx = x^{2} + C}}$

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

$:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \displaystyle\int \sf x^{2}dx + \displaystyle\int \sf2xdx}$

$:\implies \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \dfrac{x^{3}}{3} + x^{2} + C }$

$\underline{\boxed{ \sf{\displaystyle\int \sf(x^{2} + 2x)dx = \dfrac{x^{3}}{3} + x^{2} + C}}}$

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