Integration Techniques and Substitution

The video tutorial demonstrates how to integrate cosine squared of 2x. Initially, a u-substitution is considered but found unsuitable due to the absence of a sine factor in the integrand. Instead, a power reducing formula is applied, which involves doubling the angle. The integration is then split into two parts, requiring another substitution for cosine of 4x. The process concludes with the integration of each part, resulting in the final antiderivative expression.

What is the initial substitution considered for integrating cosine squared 2x?

• Let u = cos 2x

• Let u = cos^2 2x

• Let u = tan 2x

• Let u = sin 2x

Why does the initial u-substitution fail for integrating cosine squared 2x?

• The integrand contains a factor of sin 2x

• The integrand does not contain a factor of sin 2x

• The substitution is not valid for trigonometric functions

• The differential does not match the integrand

What formula is introduced to help solve the integration problem?

• Power-reducing formula

• Sum-to-product formula

• Double angle formula

• Product-to-sum formula

What is the result of applying the power-reducing formula to cosine squared 2x?

• 1/2 * (1 + cos 4x)

• 1/2 * (1 - cos 4x)

• 1/2 * (1 + sin 4x)

• 1/2 * (1 - sin 4x)

How is the integral of cosine 4x set up for further integration?

• As a single integral

• As two separate integrals

• As a product of integrals

• As a sum of integrals

What substitution is used for integrating cosine 4x?

• Let u = 2x

• Let u = 4x

• Let u = x

• Let u = 8x

What is the differential du equal to when u = 4x?

• 4 dx

• 1/4 dx

• dx

• 2 dx

What is the integral of 1 with respect to x?

• x + C

• C

• 1 + C

• 0 + C

What is the antiderivative of cosine u?

• -cos u + C

• -sin u + C

• cos u + C

• sin u + C

What is the final antiderivative obtained in the problem?

• 12x + 1/2 sin 4x + C

• 12x + sin 4x + C

• 12x + 1/4 sin 4x + C

• 12x + 1/8 sin 4x + C