Evaluate the integral
$\int x^4\left(x^5-4\right)^5 d x$
by making the substitution $u=x^5-4$.
$\square$ +C
NOTE: Your answer should be in terms of $x$ and not $u$.
Answer (1)
Perform u-substitution: Let u = x 5 − 4 , then d x d u = 5 x 4 , so d x = 5 x 4 d u .
Substitute into the integral: ∫ x 4 ( x 5 − 4 ) 5 d x = ∫ x 4 u 5 5 x 4 d u = 5 1 ∫ u 5 d u .
Integrate with respect to u: 5 1 ∫ u 5 d u = 5 1 ⋅ 6 u 6 + C = 30 u 6 + C .
Substitute back u = x 5 − 4 : 30 ( x 5 − 4 ) 6 + C . The final answer is 30 ( x 5 − 4 ) 6 .
Explanation
Problem Analysis We are asked to evaluate the integral ∫ x 4 ( x 5 − 4 ) 5 d x using the substitution u = x 5 − 4 . This is a classic u-substitution problem, where we replace a part of the integrand with a new variable to simplify the integral.
U-Substitution Let's perform the substitution u = x 5 − 4 . Then, we need to find the derivative of u with respect to x , which is d x d u = 5 x 4 . From this, we can express d x in terms of d u as d x = 5 x 4 d u .
Substitution into Integral Now, we substitute u and d x into the original integral:
Simplified Integral ∫ x 4 ( x 5 − 4 ) 5 d x = ∫ x 4 u 5 5 x 4 d u = ∫ 5 1 u 5 d u
Integration Now we can easily integrate with respect to u :
Result in terms of u 5 1 ∫ u 5 d u = 5 1 ⋅ 6 u 6 + C = 30 u 6 + C
Back to x Finally, we substitute back u = x 5 − 4 to express the result in terms of x :
Final Answer 30 ( x 5 − 4 ) 6 + C
Examples
U-substitution is a powerful technique used in many areas, such as physics. For example, when calculating the total charge of a non-uniformly charged sphere, you might need to integrate a charge density function. If the charge density depends on a complicated expression, u-substitution can simplify the integral and allow you to find the total charge more easily. This technique is also used in engineering to solve differential equations that model various physical systems.
