Question

85. Another form of ∫ sec x dxa. Verify the identity:sec x = cos x / (1 - sin² x)b. Use the identity in part (a) to verify that:∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

Accepted Answer

Step 1: Start by verifying the identity given in part (a): $$ \sec x = \frac{\cos x}{1 - \sin^2 x} . $$Recall the Pythagorean identity $$ \sin^2 x + \cos^2 x = 1 $$, and use it to rewrite the denominator $$ 1 - \sin^2 x  as$$ $$ \cos^2 x . $$Step 2: Substitute $$ 1 - \sin^2 x = \cos^2 x $$ into the right-hand side of the identity to get $$ \frac{\cos x}{\cos^2 x} = \frac{1}{\cos x} $$, which is exactly $$ \sec x . $$This completes the verification of the identity in part (a). Step 3: For part (b), start with the integral $$ \int \sec x \, dx $$ and use the identity from part (a) to rewrite $$ \sec x  as$$ $$ \frac{\cos x}{1 - \sin^2 x} . $$This transforms the integral into $$ \int \frac{\cos x}{1 - \sin^2 x} \, dx . $$Step 4: Use the substitution $$ u = \sin x $$, which implies $$ du = \cos x \, dx . $$This substitution simplifies the integral to $$ \int \frac{1}{1 - u^2} \, du . $$Step 5: Recognize that $$ \frac{1}{1 - u^2} $$ can be decomposed using partial fractions into $$ \frac{1}{(1 - u)(1 + u)} . $$Integrate the resulting expression to obtain $$ \frac{1}{2} \ln \left| \frac{1 + u}{1 - u} \right| + C . $$Finally, substitute back $$ u = \sin x  to $$complete the verification.

85. Another form of ∫ sec x dx
a. Verify the identity:
sec x = cos x / (1 - sin² x)
b. Use the identity in part (a) to verify that:
∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

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Key Concepts

Trigonometric Identities

Integration of Trigonometric Functions

Logarithmic Integration Results