Evaluate the indefinite integral.
$\int \frac{x^4}{\sqrt{1-x^{10}}} d x$
Answer (1)
Recognize the integral ∫ 1 − x 10 x 4 d x and prepare for u-substitution.
Substitute u = x 5 , which gives d u = 5 x 4 d x , and rewrite the integral in terms of u .
Evaluate the integral as 5 1 arcsin ( u ) + C .
Substitute back x 5 for u to get the final answer: 5 1 arcsin ( x 5 ) + C .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral:
Integral Expression ∫ 1 − x 10 x 4 d x
U-Substitution Setup This integral can be solved using u-substitution. Let's set:
Define u u = x 5
Find du/dx Then, the derivative of u with respect to x is:
Derivative Calculation d x d u = 5 x 4
Solve for dx This implies:
Differential du d u = 5 x 4 d x
Isolate x^4 dx So, we can express x 4 d x as:
Express x^4 dx in terms of du x 4 d x = 5 1 d u
Perform Substitution Now, substitute u = x 5 and x 4 d x = 5 1 d u into the original integral:
Transformed Integral ∫ 1 − x 10 x 4 d x = ∫ 1 − ( x 5 ) 2 1 x 4 d x = ∫ 1 − u 2 1 ⋅ 5 1 d u
Constant Factor = 5 1 ∫ 1 − u 2 1 d u
Standard Integral The integral ∫ 1 − u 2 1 d u is a standard integral, which evaluates to arcsin ( u ) + C , where C is the constant of integration. Therefore, we have:
Integrate with respect to u 5 1 ∫ 1 − u 2 1 d u = 5 1 arcsin ( u ) + C
Substitute Back Finally, substitute back u = x 5 to express the result in terms of x :
Final Result in terms of x 5 1 arcsin ( u ) + C = 5 1 arcsin ( x 5 ) + C
Conclusion Thus, the indefinite integral is:
Final Answer ∫ 1 − x 10 x 4 d x = 5 1 arcsin ( x 5 ) + C
Examples
Imagine you're designing a curved ramp for a skateboard park. The shape of the ramp can be modeled using mathematical functions, and calculating integrals like this helps determine the arc length or surface area of the ramp. This is crucial for estimating the materials needed and ensuring the ramp's structural integrity and smooth transitions for skateboarders.
