6. True or False If the substitution \( u=x^{2}+3 \) is used with \( \int_{0}^{1} x\left(x^{2}+3\right)^{3} d x \), then \( \int_{0}^{1} x\left(x^{2}+3\right)^{3} d x=\frac{1}{2} \int_{0}^{1} u^{3} d u \) 7. Multiple Choice \( \int_{-4}^{4} x^{3} d x=[(\mathbf{a}) 128 \), (b) 4 , (c) 0 , (d) 64\( ] \) 8. True or False \( \int_{0}^{5} x^{2} d x=\frac{1}{2} \int_{-5}^{5} x^{2} d x \)

6. We are given \[ \int_{0}^{1} x\left(x^{2}+3\right)^{3} d x \] and the substitution \[ u=x^{2}+3. \] Then, \[ \frac{du}{dx}=2x\quad\Longrightarrow\quad du=2x\,dx\quad\Longrightarrow\quad x\,dx=\frac{du}{2}. \] Changing the limits of integration: when \( x=0 \), \[ u=0^{2}+3=3, \] and when \( x=1 \), \[ u=1^{2}+3=4. \] Thus, the transformed integral is \[ \int_{x=0}^{x=1} x\left(x^{2}+3\right)^{3} dx=\frac{1}{2}\int_{u=3}^{u=4}u^{3}du. \] The statement claimed that the substitution leads to \[ \frac{1}{2}\int_{0}^{1} u^{3}du, \] which is incorrect because the new limits are \(3\) and \(4\), not \(0\) and \(1\). Therefore, the answer is **False**. 7. Evaluate \[ \int_{-4}^{4} x^{3} dx. \] Notice that \(x^{3}\) is an odd function; that is, \[ x^3 = -(-x)^3. \] For any odd function integrated over an interval symmetric about the origin, the integral is zero. Hence, \[ \int_{-4}^{4} x^{3} dx=0. \] The correct answer is **(c) 0**. 8. We need to decide if \[ \int_{0}^{5} x^{2} dx=\frac{1}{2}\int_{-5}^{5} x^{2} dx. \] The function \(x^2\) is even, meaning \(x^2=(-x)^2\). For any even function, \[ \int_{-5}^{5} x^{2} dx=2\int_{0}^{5} x^{2} dx. \] Dividing both sides by 2 gives \[ \int_{0}^{5} x^{2} dx=\frac{1}{2}\int_{-5}^{5} x^{2} dx, \] so the statement is **True**.