Evaluate the integral int (6x^2+10x)e^(6x) dx | Calculus 2

Our goal in Problem 1 is to find the antiderivative of the function 6 x square + 10 x times Exponential of 6 x.
Our aim in this problem is to find the antiderivative of the function 6 x square + 10 x times Exponential of 6 x.
Let us integrate the function 6 x square + 10 x times Exponential of 6 x.
observe that, at first glance we easily see that the function we have to integrate is a product of a nonlinear function, Exponential of 6 x, with a polynomial function 6 x square + 10 x.
often, when you are faced with this type of complex integration problem, you have to adopt the appropriate strategy that will help you simplify the problem, and find the antiderivative efficiently.
The two main techniques that we learn in calculus 2, for solving integrals that involve the products or quotients of nonlinear functions are the U Substitution method and the method of integration by parts.
The crucial first step of our strategy is to decide which of these two classical methods we should use.
In this case, we have a product of a nonlinear function with a polynomial function, so the best method to use is the method of integration by parts. Because this method will help us reduce the complexity of the problem by reducing the degree of the polynomial
it comes that, The polynomial function 6 x square + 10 x is of degree 2.
In sum, our first goal will be to reduce the degree of the polynomial function to 1, by using the method of integration by parts.
Let us briefly review the method of Integration by parts.
We know that the integral of u d v is equal to. u times v, minus the integral of v d u.
Consequently, we set u equal to 6 x square + 10 x, so d u equals 12 x + 10 d x.
And, d v equals to Exponential of 6 x d x, and v equals to { 1 } over { 6 } times Exponential of 6 x.
as a result, The integral of 6 x square + 10 x times Exponential of 6 x d x is equal to.
{ 6 x square + 10 x } over { 6 } times Exponential of 6 x, plus the integral of {, minus 12 x minus 10 } over { 6 } times Exponential of 6 x dx. Consequently we still have to integrate {, minus 12 x minus 10 } over { 6 } times Exponential of 6 x, using integration by parts.
Let us integrate the function {, minus 12 x minus 10 } over { 6 } times Exponential of 6 x.
Given that the function to integrate {, minus 12 x minus 10 } over { 6 } times Exponential of 6 x is still the product of a nonlinear function with a polynomial function,
we still have to use the method of integration by parts.
The main aim of these repeated utilization of the method of integration by parts is to reduced the degree of the polynomial function {, minus 12 x minus 10 } over { 6 }.
In sum, we set u equal to {, minus 12 x minus 10 } over { 6 }, so d u equals, minus 2 d x.
And, d v equals to Exponential of 6 x d x, and v equals to { 1 } over { 6 } times Exponential of 6 x.
In sum, The integral of {, minus 12 x minus 10 } over { 6 } times Exponential of 6 x d x is equal to.
{, minus 12 x minus 10 } over { 36 } times Exponential of 6 x, plus { 2 } over { 36 } times Exponential of 6 x, plus a constant C.
Now we have to aggregate the results of these repeated applications of the integration by parts to get the final value of the anti derivative of 6 x square + 10 x times Exponential of 6 x
The first integration by parts gave the following result:
So, The integral of 6 x square + 10 x times Exponential of 6 x d x is equal to.
{ 6 x square + 10 x } over { 6 } times Exponential of 6 x.
plus the integral of { 2 } over { 6 } times Exponential of 6 x d x.
From the second integration by parts, we obtained the following result:
So, The integral of { 2 } over { 6 } times Exponential of 6 x d x is equal to.
{, minus 12 x minus 10 } over { 36 } times Exponential of 6 x.
plus { 2 } over { 36 } times Exponential of 6 x plus a constant of integration C.
lastly, combining the last two results we see that, The integral of 6 x square + 10 x times Exponential of 6 x d x is equal to.
{ 6 x square + 10 x } over { 6 } times Exponential of 6 x.
plus {, minus 12 x minus 10 } over { 36 } times Exponential of 6 x.
plus { 2 } over { 36 } times Exponential of 6 x + a constant of integration C