Question

Find the value of \(\int x^3 e^{x^4} dx\).

Hint:

When using substitution, always look for a function-derivative pair. In integrals involving \(e\), a good first guess is to substitute for the exponent. Check if its derivative (or something close to it) is present as a factor in the integrand.

Answer 1

Step 1: Understanding the Concept:
This is an indefinite integral that can be solved using the method of substitution (u-substitution). We look for a part of the integrand whose derivative is also present (up to a constant factor).
Step 2: Key Formula or Approach:
1. Identify a suitable substitution, \(u\). Let \(u = x^4\).
2. Differentiate \(u\) with respect to \(x\) to find \(du\).
3. Rewrite the integral entirely in terms of \(u\) and \(du\).
4. Integrate with respect to \(u\).
5. Substitute the original expression for \(u\) back into the result.
Step 3: Detailed Explanation or Calculation:
Let the integral be \(I = \int x^3 e^{x^4} dx\).
1. Substitution:
Notice that the derivative of the exponent \(x^4\) is \(4x^3\), which is a constant multiple of the other factor \(x^3\) in the integrand. So, we choose: Let \(u = x^4\).
2. Differentiate:
Then, \(du = 4x^3 dx\).
This implies \(x^3 dx = \frac{1}{4} du\).
3. Rewrite the integral:
Substitute \(u\) and \(\frac{1}{4} du\) into the integral: \[ I = \int e^{x^4} (x^3 dx) = \int e^u \left(\frac{1}{4} du\right) \] \[ I = \frac{1}{4} \int e^u du \] 4. Integrate:
The integral of \(e^u\) is \(e^u\). \[ I = \frac{1}{4} e^u + C \] 5. Substitute back:
Replace \(u\) with \(x^4\): \[ I = \frac{1}{4} e^{x^4} + C \] Step 4: Final Answer:
The value of the integral is \(\frac{1}{4} e^{x^4} + C\), where C is the constant of integration.