   Chapter 7.1, Problem 53E

Chapter
Section
Textbook Problem

# Use integration by parts to prove the reduction formula.53. ∫ tan n x   d x   =   tan n − 1 x n   −   1   −   ∫ tan n − 2 x   d x   ( n ≠ 1 )

To determine

To prove: the reduction formula by using integration by parts.

Explanation

Given information:

The integral function is tannxdx=tann1xn1tann2xdx_.

Calculation:

Show the integral function as shown below:

tannxdx (1)

Since n is not equal to 1.

Modify Equation (1).

tannxdx=tann2xtan2xdx=tann2x(sec2x1)dx=tann2xsec2xdxtann2xdx (2)

Show the method of integration by parts as shown in below:

udv=uvvdu (3)

Consider the function u=tann2x (4)

Differentiate both sides of the Equation (4).

du=(n2)tann3xsec2xdx

Consider dv=sec2xdx

Integrate both sides of the Equation.

v=tanx

Apply method of integration by parts as shown below.

Substitute tann2x for u, sec2xdx for dv, tanx for v, and (n2)tann3xsec2xdx for du in Equation (2)

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