Given that $u=3-2 x^2$, which integral is equivalent to $\int_{-1}^0 x^3 \sqrt{3-2 x^2} d x $?
a $\quad \int_{-1}^0(3-u) \sqrt{u} d u$
b $\frac{1}{8} \int_1^3(3-u) \sqrt{u} d u$
c $\frac{1}{4} \int_{-1}^0(3-u) \sqrt{u} d u$
d $\frac{1}{2} \int_1^3(3-u) \sqrt{u} d u$
e $-\frac{1}{8} \int_1^3(3-u) \sqrt{u} d u$
Answer (1)
Find d u in terms of d x : d u = − 4 x d x .
Express x 2 in terms of u : x 2 = 2 3 − u .
Rewrite the integral: ∫ − 1 0 x 3 3 − 2 x 2 d x = ∫ 1 3 − 8 1 ( 3 − u ) u d u .
The equivalent integral is − 8 1 ∫ 1 3 ( 3 − u ) u d u .
Explanation
Problem Analysis We are given the integral ∫ − 1 0 x 3 3 − 2 x 2 d x and the substitution u = 3 − 2 x 2 . Our goal is to find the equivalent integral in terms of u .
Finding du First, we need to find the differential d u in terms of d x . Differentiating u = 3 − 2 x 2 with respect to x , we get d x d u = − 4 x
d u = − 4 x d x
Expressing x^2 in terms of u Next, we need to express x 2 in terms of u . From the substitution u = 3 − 2 x 2 , we can write 2 x 2 = 3 − u
x 2 = 2 3 − u
Rewriting the Integral Now we can rewrite the integral. We have x 3 3 − 2 x 2 d x = x 2 ⋅ x 3 − 2 x 2 d x . Substituting x 2 = 2 3 − u and d u = − 4 x d x , we get x d x = − 4 1 d u . Thus, x 3 3 − 2 x 2 d x = 2 3 − u u ( − 4 1 ) d u = − 8 1 ( 3 − u ) u d u
Changing Limits of Integration We also need to change the limits of integration. When x = − 1 , u = 3 − 2 ( − 1 ) 2 = 3 − 2 = 1 . When x = 0 , u = 3 − 2 ( 0 ) 2 = 3 . So the new limits of integration are from 1 to 3 .
Therefore, the integral becomes ∫ − 1 0 x 3 3 − 2 x 2 d x = ∫ 1 3 − 8 1 ( 3 − u ) u d u = − 8 1 ∫ 1 3 ( 3 − u ) u d u
Final Answer Comparing our result with the given options, we see that the equivalent integral is − 8 1 ∫ 1 3 ( 3 − u ) u d u
Examples
This type of substitution is often used in physics to simplify integrals that arise in problems involving energy or momentum. For example, when calculating the moment of inertia of a complex object, a suitable substitution can make the integral much easier to solve. Similarly, in signal processing, integrals involving complicated functions can be simplified using appropriate substitutions to analyze signal characteristics.
