Evaluate the Integral Using the Given Substitution $\int \sqrt { x } \cos ^ { 2 } \left( x ^ { 3 / 2 } - 6 \right) d x , \quad u = x ^ { 3 / 2 } - 6$

question 93

Multiple Choice

Evaluate the integral using the given substitution.
- $\int \sqrt { x } \cos ^ { 2 } \left( x ^ { 3 / 2 } - 6 \right) d x , \quad u = x ^ { 3 / 2 } - 6$

• A) \(\frac { 1 } { 3 } \left( x ^ { 3 / 2 } - 6 \right) + \frac { 1 } { 6 } \sin 2 \left( x ^ { 3 / 2 } - 6 \right) + C\)
• B) \(\frac { 1 } { 3 } ( \sqrt { x } ) \sin \left( x ^ { 3 / 2 } - 6 \right) + C\)
• C) \(x ^ { 3 / 2 } - 6 + \frac { 1 } { 2 } \sin 2 \left( x ^ { 3 / 2 } - 6 \right) + C\)
• D) \(\frac { 2 } { 9 } \sin ^ { 3 } \left( x ^ { 3 / 2 } - 6 \right) + C\)

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Definitions:

Mechanical Rectifier

A device that converts alternating current (AC) into direct current (DC) through mechanical means, historically used before electronic rectifiers became common.

AC

Alternating current, a type of electrical current where the flow of electrons periodically reverses direction.

DC

Direct Current, characterized by the unidirectional flow of electric charge, typically used in batteries, solar cells, and as the output of rectifiers.

Armature

The rotating coil or coils of a dynamo or electric motor, which induce electromotive force by rotating in a magnetic field.