   Chapter 7.1, Problem 53E

Chapter
Section
Textbook Problem

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

To determine

To prove: The given reduction formula using the technique of integration by parts.

Explanation

Proof:

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts in terms of u and v is given by

udv=uvvdu

Given:

The reduction formula, tannxdx=tann1xn1tann2xdx    (n1).

Calculation:

The integration on the left-hand side of the equation can be rewritten as:

tannxdx=tann2xtan2xdx

Use the identity sec2xtan2x=1 to substitute for tan2x

tannxdx=tann2x(sec2x1)dx=tann2xsec2xdxtann2xdx …… (1)

Use integration by parts to solve the integral tann2xsec2xdx. Make the choice for u and dv such that the resulting integral is easier to integrate

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