If `intx^5e^(-4x^3)dx=(1)/(48)e^(-4x^3)(f(x))+c`, where c is contant of intergration then `f(x)` equals to (a) `-4x^3-1` (b) `-1-2x^3` (c) `4x^3+1` (d) `1-2x^3`

Options

A. `-4x^3-1`
B. `4x^3+1`
C. `-2x^3-1`
D. `-2x^3+1`

Answer

To solve the integral \( \int x^5 e^{-4x^3} \, dx \) and find the function \( f(x) \) such that

\[ \int x^5 e^{-4x^3} \, dx = \frac{1}{48} e^{-4x^3} f(x) + c, \]

we will use the method of substitution.

### Step-by-Step Solution:

1. **Substitution**: Let \( t = -4x^3 \). Then, we differentiate to find \( dt \):
\[ dt = -12x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{-12x^2}. \]
We also express \( x^2 \) in terms of \( t \):
\[ x^3 = -\frac{t}{4} \quad \Rightarrow \quad x = \left(-\frac{t}{4}\right)^{1/3}. \]

2. **Express \( x^5 \)**: We can express \( x^5 \) as:
\[ x^5 = (x^3)^{5/3} = \left(-\frac{t}{4}\right)^{5/3}. \]

3. **Substituting into the integral**: Substitute \( x^5 \) and \( dx \) into the integral:
\[ \int x^5 e^{-4x^3} \, dx = \int \left(-\frac{t}{4}\right)^{5/3} e^{t} \left(\frac{dt}{-12x^2}\right). \]
We need to express \( x^2 \) in terms of \( t \):
\[ x^2 = \left(-\frac{t}{4}\right)^{2/3}. \]
Thus, we can rewrite \( dx \) as:
\[ dx = \frac{dt}{-12 \left(-\frac{t}{4}\right)^{2/3}}. \]

4. **Simplifying the integral**: The integral becomes:
\[ \int \left(-\frac{t}{4}\right)^{5/3} e^{t} \left(\frac{dt}{-12 \left(-\frac{t}{4}\right)^{2/3}}\right). \]
This simplifies to:
\[ \int \frac{t^{5/3}}{4^{5/3}} e^{t} \cdot \frac{dt}{-12 \cdot \frac{t^{2/3}}{4^{2/3}}} = \int \frac{t^{5/3}}{-12 \cdot 4^{5/3 - 2/3}} e^{t} \, dt. \]

5. **Integrating by parts**: We can apply integration by parts \( \int u \, dv = uv - \int v \, du \) where \( u = t \) and \( dv = e^{t} dt \).

6. **Final expression**: After performing the integration and substituting back \( t = -4x^3 \), we find:
\[ \int x^5 e^{-4x^3} \, dx = \frac{1}{48} e^{-4x^3} (t - 1) + c = \frac{1}{48} e^{-4x^3} (-4x^3 - 1) + c. \]

7. **Identifying \( f(x) \)**: From the expression, we can identify:
\[ f(x) = -4x^3 - 1. \]

### Conclusion:
Thus, the function \( f(x) \) is:

\[ \boxed{-4x^3 - 1}. \]