Question

How do you find the general indefinite integral of \[\left( {13{x^2} + 12{x^{ - 2}}} \right)dx\]?

Answer 1

Hint:The above question is based on the concept of integration. Since it is an indefinite integral which has no upper and lower limits, we can apply integration properties by integrating it where the power increases by one and we can find the antiderivative of the above expression.

Complete step by step answer:
Integration is a way of finding the antiderivative of any function. It is the inverse of differentiation. It denotes the summation of discrete data. Calculation of small problems is an easy task but for adding big problems which include higher limits, integration method is used. The above given expression is an indefinite integral which means there are no upper or lower limits given. The above equation should be in the below form.
\[\int {f\left( x \right) = F(x) + C} \]
where C is constant.
Now by using the property of integral,
\[\int {{{\left[ {f\left( x \right)} \right]}^n}f'\left( x \right)dx = \dfrac{{{{\left[ {f\left( x \right)} \right]}^{n + 1}}}}{{n + 1}} + C} \]
Now by applying the above given integral property we get,
\[\int {\left( {13{x^2} + 12{x^{ - 2}}} \right)dx = 13\dfrac{{{x^{2 + 1}}}}{{2 + 1}} + 12\dfrac{{{x^{ - 2 + !}}}}{{ - 2 + 1}} + c} \]
The power of the variable x is 2 which is incremented by one and becomes 3 in the first term when integrated also the integrated value is taken in the denominator.

Therefore, we can write it as \[\dfrac{{13}}{3}{x^3} - 12{x^{ - 1}} + c\]. Hence, we get the above solution for the expression.

Note:An important thing to note is that the power in the second term is negative i.e., the number is -2.The negative number will be incremented by one i.e.,-2+1 and will give the value as -1.This is the integrated value which is also written in denominator after integrating it.