Question

Integrate with respect to \[t\]
\[\int {\left( {4\cos t + {t^2}} \right)dt} \]
A) \[4\sin t + \dfrac{{{t^3}}}{3}\]
B) \[4\sin t - \dfrac{{{t^3}}}{3}\]
C) \[4\sin t + 2t\]
D) \[4\sin t + {t^3}\]

Answer 1

Hint:
Here we will use the distributive property of integration. Then we will integrate each term of the expression individually using the basic formula of integration. After simplifying the terms, we will get the required answer and hence the required integration of the given function.

Formula used: The basic formulas of the integration which we will use here are:-
1) \[\int {\cos xdx} = \sin x\]
2) \[\int {{x^2}dx} = \dfrac{{{x^3}}}{3}\]

Complete step by step solution:
Here we need to find the integration of the given expression \[\int {\left( {4\cos t + {t^2}} \right)dt} \] with respect to the given variable i.e. with respect to \[t\].
As the integrand is the sum of two functions, so we use the distributive property to integrate both the terms separately.
Now, applying the distributive property of integration. Therefore, we get
\[\int {\left( {4\cos t + {t^2}} \right)dt} = \int {4\cos tdt} + \int {{t^2}dt} \]
We will now take the constant term i.e. 4 out of the integration. So, we get
\[ \Rightarrow \int {\left( {4\cos t + {t^2}} \right)dt} = 4\int {\cos tdt} + \int {{t^2}dt} \]
Now, we will integrate the function using the basic formulas of the integration.
Applying formulas of integration \[\int {\cos xdx} = \sin x\] and \[\int {{x^2}dx} = \dfrac{{{x^3}}}{3}\] , we get
 \[ \Rightarrow \int {\left( {4\cos t + {t^2}} \right)dt} = 4\sin t + \dfrac{{{t^3}}}{3} + c\]
Here, \[c\] is the constant of integration.

Hence, the correct option is option A.

Note:
The integration function denotes the summation of data which are discrete. The integral is used to calculate the area, volume, displacement described by the function, which occurs due to a collection of small data, which cannot be measured singularly. Integration is an inverse function of differentiation. Here we have used the distributive property of integration. The distributive property of integration is used when we have to integrate the function which consists of summation or difference of different variables.