Evaluate using substitution ∫(2x^5+6x)^3(5x^4+3)dx where b=0 and a=-1
I got to here and then got stuck
u=5x^4+3
du=x^5+3x+C
∫(2x^5+6x)^3 u 1/(x^5+3x+C)
∫(x^5+3x)^3 u
du=x^5+3x+C ?? what is that all abot
You add a constant to integral is no constraints on it. BUT NOT to the derivative
if
u = 5 x^4 + 3
then
du = 20 x^3 dx period, the end but that is not what you want to do anyway
lets' try
u = 2 x^5 + 6 x
then
du = (10 x^4 + 6)dx = 2 (5 x^4+3)dx
that looks more promising :)
∫(2x^5+6x)^3(5x^4+3)dx
becomes
∫(u)^3 du/2
= (1/2) ∫u^3 du
BUT remember to change b = -1 to
2(-1)^5 +6(-6)
=-2 -36
= -38
To evaluate the integral using substitution, we can start by letting u = 2x^5 + 6x. Then, we need to find du/dx to determine the differential substitution, which will help us transform the integral from dx to du.
Let's find du/dx:
Given u = 2x^5 + 6x,
Differentiating both sides with respect to x:
du/dx = d/dx(2x^5 + 6x)
du/dx = (10x^4 + 6)
Now we can rearrange the equation to solve for dx:
du = (10x^4 + 6) dx
dx = du / (10x^4 + 6)
Next, we can substitute u and dx back into the original integral expression:
∫[(2x^5 + 6x)^3 (5x^4 + 3)] dx
Substituting u = 2x^5 + 6x and dx = du / (10x^4 + 6), the integral becomes:
∫[u^3 * (5x^4 + 3)] (du / (10x^4 + 6))
Now we can simplify the integral by factoring out the constant term:
∫[(5x^4 + 3) * u^3] (du / (10x^4 + 6))
Since we have expressed the original integral in terms of u and du, we can now determine the limits of integration, with a = -1 and b = 0.
Evaluating the integral with the given limits, we get:
∫[(5x^4 + 3) * u^3] (du / (10x^4 + 6))
Integrating from a = -1 to b = 0:
∫[(5x^4 + 3) * u^3] (du / (10x^4 + 6)) evaluated from -1 to 0
Plugging in the limits, the integral becomes:
∫[(5x^4 + 3) * u^3] (du / (10x^4 + 6)) from -1 to 0