Problem 16 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int x^{3} e^{x^{4}} d x $$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{1}{4} e^{x^4} + C \).

Step 1: Identify the Substitution

This integral can be solved using substitution. Notice the function inside the integral is composed: the exponent of the exponential term is a function of the variable of integration. We can set the inside function of the exponent as a new variable. Let \( u = x^4 \). Then, we find its differential: \( du = 4x^3 dx \). This substitution will help simplify the integral.

Step 2: Solve for dx

We need to express \( dx \) in terms of \( du \) and \( x \) so we can change the variable in our integral. From \( du = 4x^3 dx \), we can solve for \( dx \) as follows:\[ dx = \frac{du}{4x^3} \].

Step 3: Substitute and Simplify

Substitute \( u = x^4 \) and \( dx = \frac{du}{4x^3} \) into the integral. Notice \( x^3 \) in the original integral cancels with \( x^3 \) from \( dx \):\[ \int x^3 e^{x^4} dx = \int x^3 \cdot e^u \cdot \frac{du}{4x^3} = \frac{1}{4} \int e^u du \].

Step 4: Integrate with Respect to u

Now integrate \( \frac{1}{4} \int e^u du \). The integral of \( e^u \) with respect to \( u \) is \( e^u \), so:\[ \frac{1}{4} \int e^u du = \frac{1}{4} e^u + C \].

Step 5: Back-Substitute for x

Now substitute back \( u = x^4 \) to express the antiderivative in terms of \( x \):\[ \frac{1}{4} e^{x^4} + C \]. This is the function that when differentiated, should give the original integrand.

Step 6: Verify by Differentiating

Differentiate the antiderivative \( \frac{1}{4} e^{x^4} + C \) with respect to \( x \). Apply the chain rule: the derivative of \( e^{x^4} \) is \( e^{x^4} \cdot 4x^3 \).\[ \frac{d}{dx} \left( \frac{1}{4} e^{x^4} \right) = \frac{1}{4} \cdot e^{x^4} \cdot 4x^3 = x^3 e^{x^4} \]. This matches the original integrand, confirming our solution is correct.

Key Concepts

Substitution Method

The Substitution Method is a powerful tool in integration, particularly useful when dealing with composite functions. In the given exercise, we face the challenge of integrating a complex exponential function. The trick is to simplify it by substituting a part of the function with a new variable.

Given the integral \( \int x^3 e^{x^4} dx \), we notice the exponent \( x^4 \) as the inner function. By setting \( u = x^4 \), we target this complexity for simplification. But don't stop there; we also need to transform the differential, which is found as \( du = 4x^3 dx \).

Here's how you can approach substitution:

• Identify the inner function and set it as \( u \).
• Compute \( du \) to reflect the change in the differential.
• Substitute both \( u \) and \( dx \) back into the integral, simplifying wherever possible.

This helps in transforming an intimidating integral into a more manageable form, making use of basic integral rules.

Exponential Integration

When you encounter an exponential function within an integral, it's crucial to recall the simplicity of integrating exponential expressions. The exponential function \( e^x \) is distinct because it has the same form when integrated as when differentiated.

In our substituted integral \( \frac{1}{4} \int e^u du \), we leverage this property. The integral of \( e^u \) is simply \( e^u \) itself. Thus, the solution becomes:

• Integrate \( e^u \) to get \( e^u \).
• Don't forget to incorporate the scale factor, here \( \frac{1}{4} \), into the integral.
• Reincorporate back the constant \( C \), which represents an indefinite integral.

The result, \( \frac{1}{4} e^u + C \), emphasizes how straightforward exponential integration can be when paired with substitution.

Differentiation Verification

Verification through differentiation is a crucial step to ensure the accuracy of your integral's solution. Here, it serves as a check to confirm that our antiderivative indeed reflects the original integrand.

For our exercise, the antiderivative \( \frac{1}{4} e^{x^4} + C \) is derived, and we differentiate it to validate our work:

• Apply the chain rule: the derivative of \( e^{x^4} \) involves multiplying by the derivative of \( x^4 \), which is \( 4x^3 \).
• Thus, differentiate \( \frac{1}{4} e^{x^4} \) to get \( \frac{1}{4} \times e^{x^4} \times 4x^3 \).
• Simplify to find \( x^3 e^{x^4} \), which matches our initial integrand.

This process assures that the original function is correctly obtained from the derived expression, solidifying the integration as properly executed.