Final answer to the problem
Step-by-step Solution
Specify the solving method
Simplify $\left(e^x\right)^3$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $x$ and $n$ equals $3$
We can solve the integral $\int x^5e^{3x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $3x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
Take the constant $\frac{1}{729}$ out of the integral
Divide $1$ by $729$
We can solve the integral $\int u^5e^udu$ by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form $\int P(x)T(x) dx$. $P(x)$ is typically a polynomial function and $T(x)$ is a transcendent function such as $\sin(x)$, $\cos(x)$ and $e^x$. The first step is to choose functions $P(x)$ and $T(x)$
Derive $P(x)$ until it becomes $0$
Integrate $T(x)$ as many times as we have had to derive $P(x)$, so we must integrate $e^u$ a total of $6$ times
With the derivatives and integrals of both functions we build the following table
Then the solution is the sum of the products of the derivatives and the integrals according to the previous table. The first term consists of the product of the polynomial function by the first integral. The second term is the product of the first derivative by the second integral, and so on.
When multiplying exponents with same base we can add the exponents
When multiplying exponents with same base you can add the exponents: $u^5e^u\cdot u$
When multiplying two powers that have the same base ($u^{3}$), you can add the exponents
Combining like terms $u^{6}e^u$ and $-5u^{6}e^u$
Simplify $\left(u^{3}\right)^2$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $3$ and $n$ equals $2$
When multiplying exponents with same base we can add the exponents
When multiplying exponents with same base you can add the exponents: $120u\cdot u^{5}e^u$
Combining like terms $-4u^{6}e^u$ and $20u^{6}e^u$
Combining like terms $16u^{6}e^u$ and $-60u^{6}e^u$
Combining like terms $-44u^{6}e^u$ and $120u^{6}e^u$
Combining like terms $76u^{6}e^u$ and $-120u^{6}e^u$
Multiply $\frac{1}{729}$ times $-44$
Replace $u$ with the value that we assigned to it in the beginning: $3x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$