Indefinite Integrals and Substitution

This lesson covers evaluating indefinite integrals using the method of u-substitution. It provides two examples: the first involves the integral of sine x divided by cosine to the fourth x, and the second involves a double angle in the numerator. The lesson explains the substitution process, applying the power rule, and simplifying the results. It also discusses the use of natural logarithms in integration and the conditions under which absolute values can be omitted.

What is the primary method used in this lesson to evaluate indefinite integrals?

• Partial fraction decomposition

• U-substitution

• Trigonometric substitution

• Integration by parts

In the first example, what substitution is made for u?

• u = secant x

• u = tangent x

• u = cosine x

• u = sine x

What is the differential du when u = cosine x?

• du = -cosine x dx

• du = sine x dx

• du = cosine x dx

• du = -sine x dx

How is the integral simplified after substitution in the first example?

• By using partial fractions

• By using the chain rule

• By applying the power rule

• By using integration by parts

What is the result of the first example's integration in terms of x?

• 1/3 * cosine^3 x + C

• 1/3 * sine^3 x + C

• 1/3 * secant^3 x + C

• 1/3 * tangent^3 x + C

In the second example, what is the expression for sine 2x?

• 2 sine x cosine x

• sine^2 x + cosine^2 x

• sine x / cosine x

• cosine x / sine x

What substitution is made for u in the second example?

• u = sine x

• u = cosine x

• u = 8 + cosine^2 x

• u = 8 + sine^2 x

What is the differential du when u = 8 + cosine^2 x?

• du = -2 cosine x dx

• du = 2 cosine x dx

• du = -2 sine x dx

• du = 2 sine x dx

How is the integral simplified after substitution in the second example?

• By using natural logarithms

• By using partial fractions

• By applying the power rule

• By using the chain rule

What is the final expression for the second example's integration in terms of x?

• -2 ln |8 + sine^2 x| + C

• -2 ln |8 + cosine^2 x| + C

• -2 ln |8 + tangent^2 x| + C

• -2 ln |8 + secant^2 x| + C