If `int_(0)^(1)x^(11)e^(-x^(24))dx=A`, and `int_(0)^(1)x^(3)e^(-x^(8))dx=B`, then the relation between A and B is

Options

A. `A = 3B`
B. `B = 3A`
C. `A+3B=0`
D. `B+3A=0`

Answer

To find the relation between the integrals \( A \) and \( B \), we will evaluate each integral step by step.

### Step 1: Evaluate \( A \)

Given:
\[ A = \int_0^1 x^{11} e^{-x^{24}} \, dx \]

We will use a substitution. Let:
\[ t = x^{12} \implies dt = 12x^{11} \, dx \implies dx = \frac{dt}{12x^{11}} \]

From the substitution \( t = x^{12} \):
- When \( x = 0 \), \( t = 0^{12} = 0 \)
- When \( x = 1 \), \( t = 1^{12} = 1 \)

Now, we need to express \( x^{11} \) in terms of \( t \):
\[ x = t^{1/12} \implies x^{11} = (t^{1/12})^{11} = t^{11/12} \]

Substituting these into the integral:
\[ A = \int_0^1 t^{11/12} e^{-(t^{2})} \cdot \frac{dt}{12} = \frac{1}{12} \int_0^1 e^{-t^2} \, dt \]

### Step 2: Evaluate \( B \)

Given:
\[ B = \int_0^1 x^3 e^{-x^8} \, dx \]

We will use a similar substitution. Let:
\[ u = x^4 \implies du = 4x^3 \, dx \implies dx = \frac{du}{4x^3} \]

From the substitution \( u = x^4 \):
- When \( x = 0 \), \( u = 0^{4} = 0 \)
- When \( x = 1 \), \( u = 1^{4} = 1 \)

Now, we need to express \( x^3 \) in terms of \( u \):
\[ x = u^{1/4} \implies x^3 = (u^{1/4})^3 = u^{3/4} \]

Substituting these into the integral:
\[ B = \int_0^1 u^{3/4} e^{-(u^{2})} \cdot \frac{du}{4} = \frac{1}{4} \int_0^1 e^{-u^2} \, du \]

### Step 3: Relate \( A \) and \( B \)

Now we have:
\[ A = \frac{1}{12} \int_0^1 e^{-t^2} \, dt \]
\[ B = \frac{1}{4} \int_0^1 e^{-u^2} \, du \]

Since both integrals \( \int_0^1 e^{-t^2} \, dt \) and \( \int_0^1 e^{-u^2} \, du \) are equal, we can denote them as \( C \):
\[ A = \frac{1}{12} C \]
\[ B = \frac{1}{4} C \]

Now, we can express \( B \) in terms of \( A \):
\[ B = \frac{1}{4} C = \frac{1}{4} \cdot 12A = 3A \]

### Final Relation

Thus, the relation between \( A \) and \( B \) is:
\[ B = 3A \]