Evaluating the Definite Integral of x√(x^21) from 0 to 1

When dealing with definite integrals, the process involves determining the antiderivative of a function and evaluating it over a specific interval. In the case of the integral (int_{0}^{1} xsqrt{x^{21}} , dx), we can use the substitution method to simplify the process. This article will guide you through the steps and provide the final result.

Step-by-Step Solution Using Substitution

In this problem, we will use the substitution method to simplify the integral (int_{0}^{1} xsqrt{x^{21}} , dx).

Substitution Process

We start by letting (u x^{21}). The derivative of (u) with respect to (x) is:

(frac{du}{dx} 21x^{20})

Since we are dealing with (x) and (x^{21}), it's more straightforward to use a different substitution. Let's use (u x^2 1). This substitution can simplify the square root term and the differential element:

(u x^2 1)

Solving for (x), we get:

(x dx frac{du}{2})

Considering the limits of integration: when (x 0), (u 1), and when (x 1), (u 2). Therefore, we can rewrite the integral as:

(int_{0}^{1} xsqrt{x^2 1} , dx frac{1}{2} int_{1}^{2} sqrt{u} , du)

Evaluating the Simplified Integral

The integral (frac{1}{2} int_{1}^{2} sqrt{u} , du) is a standard form, where we can integrate (sqrt{u}) with respect to (u).

(int sqrt{u} , du int u^{1/2} , du frac{2}{3} u^{3/2} C)

Applying the limits of integration, we get:

(frac{1}{2} left[frac{2}{3} u^{3/2} right]_{1}^{2} frac{1}{3} left[2^{3/2} - 1^{3/2} right] frac{1}{3} left[2sqrt{2} - 1 right])

Final Result

Therefore, the definite integral of (xsqrt{x^2 1}) from 0 to 1 is:

(int_{0}^{1} xsqrt{x^2 1} , dx frac{1}{3} left[2sqrt{2} - 1 right])

Conclusion

The process of evaluating the definite integral of (xsqrt{x^2 1}) using the substitution method clearly demonstrates the effectiveness of this technique. The final result is (frac{1}{3} left[2sqrt{2} - 1 right]).

By understanding these methods, you can tackle more complex integrals and apply similar techniques to other problems involving algebraic manipulations and substitution.