Question

\(\int x^5e^{1-x^6}dx=\)

• A. \(\frac{1}{6}e^{1-x^6}+C\)
• B. \(-e^{1-x^6}+C\)
• C. \(\frac{-1}{6}e^{1-x^6}+C\)
• D. \(\frac{x^5}{5}e^{1-x^6}+C\)
• E. \(\frac{x^6}{6}e^{1-x^6}+C\)

Answer 1

The Correct Option is C

We are given the integral \( \int x^5 e^{1 - x^6} \, dx \) and asked to find the solution. 

We can solve this integral using substitution. Let:

\( u = 1 - x^6 \),

so that \( du = -6x^5 \, dx \), or equivalently, \( -\frac{1}{6} du = x^5 \, dx \).

Substituting into the integral, we get:

\( \int x^5 e^{1 - x^6} \, dx = -\frac{1}{6} \int e^u \, du \).

The integral of \( e^u \) is \( e^u \), so we have:

\( -\frac{1}{6} e^u + C \).

Substitute back \( u = 1 - x^6 \):

\( -\frac{1}{6} e^{1 - x^6} + C \).

The correct answer is \( \frac{-1}{6} e^{1 - x^6} + C \).