use substitution to find the indefinite integral
(x^3+3x)(x^4+6x^2+8)^9dx
To find the indefinite integral of the given function using substitution, we will follow these steps:
Step 1: Choose a suitable substitution.
Step 2: Calculate the derivative of the substitution.
Step 3: Adjust the integral in terms of the substitution.
Step 4: Solve the adjusted integral.
Step 5: Substitute back using the original variable.
Let's go through each step in detail:
Step 1: Choose a suitable substitution.
Looking at the function, we see that the term inside the parentheses, (x^4+6x^2+8), can be a good candidate for substitution. Let's substitute it with a new variable, u:
u = x^4 + 6x^2 + 8
Step 2: Calculate the derivative of the substitution.
To find the derivative of u with respect to x, we will use the chain rule:
du/dx = d/dx (x^4 + 6x^2 + 8)
= 4x^3 + 12x
Step 3: Adjust the integral in terms of the substitution.
To rewrite the integral in terms of u, we need to replace dx with du. Rearranging the equation from Step 2, we can solve for dx:
dx = du / (4x^3 + 12x)
Now, the integral becomes:
∫((x^3 + 3x)(x^4 + 6x^2 + 8)^9)dx
= ∫((x^3 + 3x)u^9) * (du / (4x^3 + 12x))
Step 4: Solve the adjusted integral.
Substituting the adjusted integral, we can simplify it further:
∫((x^3 + 3x)u^9) * (du / (4x^3 + 12x))
= 1/4 ∫((x(x^2 + 3))u^9) * (du / x(x^2 + 3))
= 1/4 ∫u^9 du
Now, the integral becomes relatively easy to solve as we have removed the variable x completely.
Step 5: Substitute back using the original variable.
Integrating u^9 with respect to u, we get:
∫u^9 du = (1/10)u^10 + C
Finally, substituting back u with x^4 + 6x^2 + 8, we arrive at the solution:
∫((x^3 + 3x)(x^4 + 6x^2 + 8)^9)dx = (1/10)(x^4 + 6x^2 + 8)^10 + C
Therefore, the indefinite integral of the given function is (1/10)(x^4 + 6x^2 + 8)^10 + C.