Question

Integrate the function in the pic pls, thankyou ​

Answer 1

The Given Integral is :-

• ${ \sf { \displaystyle { \int \limits_{0}^{2} { \bigg [ x³ + 3 } \bigg ] \: dx } } }$

Solution :-

Let us assume that ;

${ \sf { \displaystyle { I = \int \limits_{0}^{2} { \bigg [x³ + 3 } \bigg ]\: dx } } }$

We knows that ;

${ \underline { \boxed { \sf { \bf { \displaystyle { \int {x}^{n} dx = \dfrac{{x}^{n+1}}{n+1} } } } } } }$

${ \underline { \boxed { \sf { \bf { \displaystyle { \int a dx = ax } } } } } }$

For some constant " a "

Now applying this same rule and separating the terms we have ;

$: \longmapsto { \displaystyle { \sf { I = \bigg [ \int \limits_{0}^{2} { ( x³) } \: dx + \int \limits_{0}^{2} {3} \: dx } \bigg ] } }$

$: \longmapsto { \displaystyle { \sf { I = \bigg [ \left|_{\:}^{\:} {\dfrac{{x}^{3+1}}{3+1}} \right|_{0}^{2} + \left|_{\:}^{\:} {3x} \right|_{0}^{2} \bigg ] } } }$

$: \longmapsto { \displaystyle { \sf { I = \bigg [ \left|_{\:}^{\:} {\dfrac{{x}^{4}}{4}} \right|_{0}^{2} + \left|_{\:}^{\:} {3x} \right|_{0}^{2} \bigg ] } } }$

Applying the limits we have ;

${ : \longmapsto { \displaystyle { \sf { I = { \bigg [ \dfrac{2⁴}{4} - \dfrac{0⁴}{4} } \bigg ] + { \bigg [ 3 × 2 - 3 × 0 } \bigg ] } } } }$

${ : \longmapsto { \displaystyle { \sf { I = { \bigg [ \dfrac{16}{4} - 0 } \bigg ] + { \bigg [ 6 - 0 } \bigg ] } } } }$

${ : \longmapsto { \displaystyle { \sf { I = { \bigg [ 4 - 0 } \bigg ] + { \bigg [ 6 } \bigg ] } } } }$

${ : \longmapsto { \displaystyle { \sf { I = { \bigg [ 4 + 6 } \bigg ] } } } }$

${ : \longmapsto { \displaystyle { \sf { I = { \bigg [ 10 } \bigg ] } } } }$

${ : \longmapsto { \displaystyle { \sf { I = 10 } } } }$

Now , Add Constant of Integration i.e "C"

${ : \longmapsto { \displaystyle { \sf { I = 10 + C } } } }$

${ \underline { \boxed { \sf { \displaystyle { \therefore \int \limits_{0}^{2} { \bigg [ x³ + 3 } \bigg ]\: dx = 10 + C} } } } }$

Here , the given Integral Transforms to 10 + C

Henceforth , The Required Answer is 10 + C