   Chapter 7.1, Problem 54E

Chapter
Section
Textbook Problem

# Use integration by parts to prove the reduction formula. ∫ sec n x   d x = tan x   sec n − 2 x n − 1 + n − 2 n − 1 ∫ sec 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, secnxdx=tanxsecn2xn1+n2n1secn2xdx    (n1).

Calculation:

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

secnxdx=secn2xsec2xdx

Use integration by parts to solve the integration secn2xsec2xdx. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=secn2x      dv=sec2xdx

Then, the differentiation of u and antiderivative of dv will be

du=(n2)secn3xsecxtanxdx     v=tanx

Using the integration by parts formula above, the given integration will become

secn2xsec2xdx=secn2xtanxtanx(n2)secn3xsecxtanxdx=secn2xtanx(n2)tan2xsecn2xdx

Use the identity sec2xtan2x=1 to substitute for

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