Problem 59 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{\sqrt{7}} 7 x \sqrt[3]{1+x^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{315}{8} \).

Step 1: Substitution

To evaluate the integral \( \int_{0}^{\sqrt{7}} 7x \sqrt[3]{1+x^2} \, dx \), we perform a substitution. Let \( u = 1 + x^2 \). Then, \( du = 2x \, dx \). Solving for \( x \, dx \), we have \( x \, dx = \frac{1}{2} du \). Adjust the constants by taking out what is needed in the integral to match this form. The substitution gives us a new integral in terms of \( u \).

Step 2: Change of Bounds

When \( x = 0 \), \( u = 1 + 0^2 = 1 \). When \( x = \sqrt{7} \), \( u = 1 + (\sqrt{7})^2 = 8 \). Thus, the limits for \( u \) are \( 1 \) to \( 8 \).

Step 3: Substitute Back into Integral

Substitute \( u \) and \( x \, dx \) back into the integral: \[ \int_{1}^{8} 7 \left( \frac{1}{2} du \right) \sqrt[3]{u}. \] Factor the constant out of the integral: \[ \frac{7}{2} \int_{1}^{8} u^{1/3} \, du. \]

Step 4: Integrate

Calculate the antiderivative of \( u^{1/3} \):\[ \int u^{1/3} \, du = \frac{u^{4/3}}{4/3} = \frac{3}{4}u^{4/3}. \] Substitute this back into the integral:\[ \frac{7}{2} \left[ \frac{3}{4}u^{4/3} \right]_{1}^{8}. \]

Step 5: Simplify and Evaluate at Limits

Multiply the constants: \[ \frac{7}{2} \cdot \frac{3}{4} = \frac{21}{8}. \]Evaluate the integrated function at the bounds:\[ \frac{21}{8} \left( 8^{4/3} - 1^{4/3} \right). \]Calculate \( 8^{4/3} \) and \( 1^{4/3} \):- \( 8^{4/3} = (2^3)^{4/3} = 2^4 = 16 \). - \( 1^{4/3} = 1 \).Therefore, we have: \[ \frac{21}{8} (16 - 1) = \frac{21}{8} \cdot 15 = \frac{315}{8}. \]

Key Concepts

Integration by Substitution

Integration by substitution is a widely used method to simplify complex integrals by transforming them into a form that is easier to evaluate. The idea is to introduce a new variable to replace a part of the integral function, turning it into a standard form we can work with.
In our exercise, we start by letting a new variable, say \( u \,\rparenthesis\), represent a function of the original variable, \( x\). The substitution \( u = 1 + x^2 \) is crucial for simplifying the integral \( \int_{0}^{\sqrt{7}} 7x \sqrt[3]{1+x^2} \, dx \). This step helps us because the derivative \( du = 2x \, dx \) fits neatly with the \( x \, dx \) already present in the integral.
By substituting, the integral changes from a complicated expression in terms of \( x \) to a simpler one in terms of \( u \). This makes the subsequent steps more straightforward and the computation much easier to handle.

Change of Variable

The change of variable is a follow-up step to the integration by substitution, where we shift the differential bounds from the variable \( x \) to the new variable \( u \). In this process, we need to calculate the values of \( u \) that correspond to the original limits of integration when \( x = 0 \) and \( x = \sqrt{7} \).
For our example:

• When \( x = 0 \, \) we compute \( u = 1 + 0^2 = 1 \).
• When \( x = \sqrt{7} \, \) \( u = 1 + (\sqrt{7})^2 = 8 \).

Now, the integral's bounds are updated to \( u = 1 \) to \( u = 8 \), ensuring our substitution accurately mirrors the original problem. This step ensures that after transformation, the limits respect the initial conditions of the problem, maintaining the integral’s value.

Antiderivative

Finding the antiderivative is central to solving integrals as it reverses differentiation. The antiderivative of a function gives us the original function that was differentiated to produce it. In our exercise, we calculate the antiderivative of \( u^{1/3} \).
We express this as:

• \( \int u^{1/3} \, du = \frac{u^{4/3}}{4/3} = \frac{3}{4}u^{4/3} \).

This transformation simplifies the expression into a measurable form, allowing us to apply the fundamental theorem of calculus effectively.
By integrating, we change the expression back into a polynomial-like form that is ready to be evaluated at specific limits, reassembling the function the integral derives from.

Limits of Integration

The limits of integration are boundaries that define the start and end points for evaluating the definite integral. They define the interval over which the function will be integrated, thereby determining the area under the curve within that interval.
After performing the substitution and changing the variable, the limits for our new integral become \( u = 1 \) to \( u = 8 \). Applying these limits correctly is crucial for an accurate solution.
Finally, substituting back to compute:

• \( \frac{7}{2} \left[ \frac{3}{4}u^{4/3} \right]_{1}^{8} \).

Here we plug the upper limit \( u = 8 \), and the lower limit \( u = 1 \) into the antiderivative. Evaluating the difference between these values yields the final solution \( \frac{315}{8} \), representing the definite integral of the original function.