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 $$ \int_{0}^{5} x^{2} d x=\frac{1}{2} \int_{-5}^{5} x^{2} d x $$

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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**.
6. **False** 7. **(c) 0** 8. **True**

True or False If the substitution is used with , then

Multiple Choice ,
(b) 4 ,
© 0 ,
(d) 64

True or False

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True or False If the substitution is used with , then Multiple Choice , (b) 4 , © 0 , (d) 64 True or False