FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
(X^4-16)^5 X^3 DX
Hint:
note the x^4 in parentheses, and that d(x^4)/dx = 4x^3, so substituting
y=x^4 would have a good chance of success.
If
y=x^4-16,
dy=4x^3dx
So
I=∫(X^4-16)^5 X^3 DX
=∫y^5(dy/4)
proceed with the power rule and put back y=x^4-16 into I.
Do not forget the integration constant C.
To find the indefinite integral of the function (x^4 - 16)^5 * x^3 dx using the substitution method, we can follow these steps:
Step 1: Identify a suitable substitution:
Let's substitute u = x^4 - 16. Then, we can find du/dx by differentiating both sides with respect to x:
du/dx = 4x^3
Step 2: Express the function in terms of the new variable u:
We can rewrite the original function (x^4 - 16)^5 * x^3 dx in terms of u:
(x^4 - 16)^5 * x^3 dx = (u^5) * (1/4) * du
(The x^3 dx is replaced by du, and we substitute x^4 - 16 with u.)
Step 3: Rewrite the original variable in terms of the new variable:
Now, let's substitute x^4 - 16 back into our expression to rewrite the integral in terms of x:
∫ (x^4 - 16)^5 * x^3 dx = (1/4) ∫ u^5 du
Step 4: Evaluate the indefinite integral:
∫ u^5 du = (1/4) * (u^6 / 6) + C
(Integrating u^5 with respect to u gives u^6 / 6. Here, C represents the constant of integration.)
Step 5: Substitute the original variable back in:
Finally, substitute x^4 - 16 back for u:
∫ (x^4 - 16)^5 * x^3 dx = (1/4) * ((x^4 - 16)^6 / 6) + C
That is the result of finding the indefinite integral using the substitution method for the given function.