FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
�ç(X^4-16)^5 X^3 DX
To find the indefinite integral of the given expression using the substitution method, follow these steps:
Step 1: Choose an appropriate substitution.
Let's let u = X^4 - 16. Then, differentiating both sides with respect to X gives du = 4X^3 dX.
Step 2: Substitute the expression and differential.
Substituting u and du into the integral expression gives:
∫ (u^5) (X^3) dX / (4X^3)
Step 3: Simplify the expression.
We can simplify the expression by canceling out common terms:
∫ (u^5 / 4) dX
Step 4: Integrate with respect to the new variable.
Integrating the simplified expression with respect to X gives:
(1/4) ∫ u^5 dX
Step 5: Change back to the original variable.
Now, we need to change variables back to X. Recall that u = X^4 - 16. Now, solve for X in terms of u:
u = X^4 - 16
u + 16 = X^4
(X^4) = u + 16
X^4 = √(u + 16)
X = (√(u + 16))^(1/4)
Step 6: Substitute the original variable back into the integral expression.
Substituting X with (√(u + 16))^(1/4) in the integral expression gives:
(1/4) ∫ u^5 dX = (1/4) ∫ u^5 ((√(u + 16))^(1/4)) dX
Step 7: Arrange the expression for integration by substitution.
Rearranging the expression gives:
(1/4) ∫ (u^5/4) ((u + 16)^(-1/4)) dX
Step 8: Integrate using u as the variable.
Integrating the expression with respect to u gives:
(1/4) * (u^(9/4))/(9/4) * (u + 16)^(-1/4) + C
Step 9: Simplify the expression.
Simplifying the expression gives:
(4/36) * (u^(9/4)) * (u + 16)^(-1/4) + C
(u^(9/4)) * (u + 16)^(-1/4) / 9 + C
Step 10: Substitute the original variable back into the expression.
Substituting X^4 - 16 for u in the simplified expression gives:
(X^4 - 16)^(9/4) * (X^4 - 16 + 16)^(-1/4) / 9 + C
Step 11: Simplify the expression further if necessary.
If desired, you can simplify the expression further by simplifying the denominator:
(X^4 - 16)^(9/4) * (X^4)^(1/4) / 9 + C
Final Answer:
Therefore, the indefinite integral of (X^4 - 16)^5 * X^3 dX using the substitution method is:
(X^4 - 16)^(9/4) * X / 9 + C, where C is the constant of integration.