Final answer to the problem
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\cdot \frac{1}{2}e^{2\cdot 3x}-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\cdot \frac{1}{3}e^{3\cdot 3x}+C_0$
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Step-by-step Solution
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1
We can solve the integral $\int\left(3x^2+e^{3x}\right)^3dx$ 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
$u=3x$
Intermediate steps
2
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
$du=3dx$
Explain this step further
3
Isolate $dx$ in the previous equation
$du=3dx$
Intermediate steps
4
Rewriting $x$ in terms of $u$
$x=\frac{u}{3}$
Explain this step further
Intermediate steps
5
Substituting $u$, $dx$ and $x$ in the integral and simplify
$\int\frac{\left(\frac{1}{3}u^2+e^u\right)^3}{3}du$
Explain this step further
6
Take the constant $\frac{1}{3}$ out of the integral
$\frac{1}{3}\int\left(\frac{1}{3}u^2+e^u\right)^3du$
$\frac{1}{3}\int\left(\frac{1}{3}u^2+e^u\right)^3du$
Intermediate steps
8
Rewrite the integrand $\left(\frac{1}{3}u^2+e^u\right)^3$ in expanded form
$\frac{1}{3}\int\left(\frac{1}{27}u^{6}+\frac{1}{3}u^{4}e^u+u^2e^{2u}+e^{3u}\right)du$
Explain this step further
9
Expand the integral $\int\left(\frac{1}{27}u^{6}+\frac{1}{3}u^{4}e^u+u^2e^{2u}+e^{3u}\right)du$ into $4$ integrals using the sum rule for integrals, to then solve each integral separately
$\frac{1}{3}\int\frac{1}{27}u^{6}du+\frac{1}{3}\int\frac{1}{3}u^{4}e^udu+\frac{1}{3}\int u^2e^{2u}du+\frac{1}{3}\int e^{3u}du$
Intermediate steps
10
The integral $\frac{1}{3}\int\frac{1}{27}u^{6}du$ results in: $\frac{27}{7}x^{7}$
$\frac{27}{7}x^{7}$
Explain this step further
Intermediate steps
11
The integral $\frac{1}{3}\int\frac{1}{3}u^{4}e^udu$ results in: $243e^{3x}x^{5}$
$243e^{3x}x^{5}$
Explain this step further
Intermediate steps
12
The integral $\frac{1}{3}\int u^2e^{2u}du$ results in: $\frac{1}{6}e^{6x}\left(3x\right)^2-2147483648e^{2\cdot 3x}x+\frac{1}{6}\int e^{2u}du$
$\frac{1}{6}e^{6x}\left(3x\right)^2-2147483648e^{2\cdot 3x}x+\frac{1}{6}\int e^{2u}du$
Explain this step further
13
Gather the results of all integrals
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
14
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
15
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
16
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
17
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
18
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
19
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
20
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
21
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
22
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
23
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
24
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
25
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
26
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
27
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
28
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
29
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
30
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
31
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
32
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
33
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
34
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
35
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
36
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
37
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
38
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
39
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
40
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
41
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
42
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
43
Multiply $-2147483648$ times $3$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Intermediate steps
44
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
45
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
46
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
47
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
48
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
49
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
50
Simplify the expression inside the integral
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\int e^{2u}du-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\int e^{3u}du$
Explain this step further
Intermediate steps
51
The integral $\frac{1}{6}\int e^{2u}du$ results in: $\frac{1}{6}\cdot \frac{1}{2}e^{2\cdot 3x}$
$\frac{1}{6}\cdot \frac{1}{2}e^{2\cdot 3x}$
Explain this step further
Intermediate steps
52
The integral $\frac{1}{3}\int e^{3u}du$ results in: $\frac{1}{3}\cdot \frac{1}{3}e^{3\cdot 3x}$
$\frac{1}{3}\cdot \frac{1}{3}e^{3\cdot 3x}$
Explain this step further
53
Gather the results of all integrals
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\cdot \frac{1}{2}e^{2\cdot 3x}-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\cdot \frac{1}{3}e^{3\cdot 3x}$
54
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\cdot \frac{1}{2}e^{2\cdot 3x}-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\cdot \frac{1}{3}e^{3\cdot 3x}+C_0$
Final answer to the problem$\frac{27}{7}x^{7}+243e^{3x}x^{5}+\frac{1}{6}\cdot \frac{1}{2}e^{2\cdot 3x}-2147483648e^{2\cdot 3x}x+\frac{1}{6}e^{6x}\left(3x\right)^2+\frac{1}{3}\cdot \frac{1}{3}e^{3\cdot 3x}+C_0$