Question

Let $ f(x) = \int x^3 \sqrt{3-x^2} dx $. If $ 5f(\sqrt{2}) = -4 $, then $ f(1) $ is equal to

• A. \( -\frac{2\sqrt{2}}{5} \)
• B. \( -\frac{8\sqrt{2}}{5} \)
• C. \( -\frac{4\sqrt{2}}{5} \)
• D. \( -\frac{6\sqrt{2}}{5} \)

Hint:

Use substitution to simplify the integral and then use the given condition to find the constant of integration.

Answer 1

The Correct Option is D

To determine \( f(1) \), we first analyze the function \( f(x) = \int x^3 \sqrt{3-x^2} \, dx \). We are given the condition \( 5f(\sqrt{2}) = -4 \).

We evaluate the integral \( \int x^3 \sqrt{3-x^2} \, dx \) using substitution. Let \( u = 3 - x^2 \). Then \( du = -2x \, dx \), which implies \( x \, dx = -\frac{1}{2} \, du \). Since \( x^2 = 3 - u \), we have \( x^3 = x(3-u) \).

Substituting these into the integral gives:

\[ f(x) = \int x(3-u)\sqrt{u} \left(-\frac{1}{2} \right) \, du \]

This simplifies to:

\[ = -\frac{1}{2} \int (3 - u)\sqrt{u} \, du = -\frac{1}{2} \left( \int 3u^{1/2} \, du - \int u^{3/2} \, du \right) \]

Evaluating the individual integrals:

\[ \int 3u^{1/2} \, du = 3 \cdot \frac{2}{3} u^{3/2} = 2u^{3/2} \]

\[ \int u^{3/2} \, du = \frac{2}{5} u^{5/2} \]

Combining these results, we get the expression for \( f(x) \):

\[ f(x) = -\frac{1}{2} \left( 2u^{3/2} - \frac{2}{5} u^{5/2} \right) = -\frac{1}{2} \left( 2(3-x^2)^{3/2} - \frac{2}{5} (3-x^2)^{5/2} \right) \]

Now, we apply the given condition \( 5f(\sqrt{2}) = -4 \). Substituting \( x = \sqrt{2} \):

\[ 5 \left( -\frac{1}{2} \left( 2(3-(\sqrt{2})^2)^{3/2} - \frac{2}{5} (3-(\sqrt{2})^2)^{5/2} \right) \right) = -4 \]

With \( u = 3 - (\sqrt{2})^2 = 1 \), the equation becomes:

\[ 5 \left( -\frac{1}{2} \cdot \left( 2 \cdot 1^{3/2} - \frac{2}{5} \cdot 1^{5/2} \right) \right) = -4 \]

Simplifying this equation:

\[ 5 \left( -\frac{1}{2} \cdot \left(2 - \frac{2}{5} \right) \right) = -4 \]

This leads to \( f(\sqrt{2}) = \frac{4}{5} \).

Finally, we calculate \( f(1) \) by substituting \( x = 1 \) into the expression for \( f(x) \):

\[ f(1) = -\frac{1}{2} \cdot (2 \cdot (3-1)^{3/2} - \frac{2}{5} \cdot (3-1)^{5/2}) \]

With \( u = 2 \), the expression evaluates to:

\[ -\frac{1}{2} \cdot (2 \cdot 2^{3/2} - \frac{2}{5} \cdot 2^{5/2}) = -\frac{6\sqrt{2}}{5} \]

Therefore, \( f(1) = -\frac{6\sqrt{2}}{5} \).