Question

Using different substitutionsShow that the integral∫((x² - 1)(x + 1))^(-2/3) dxcan be evaluated with any of the following substitutions.c. u = arctan xWhat is the value of the integral?

Accepted Answer

Start with the integral $$ \int ((x^{2} - 1)(x + 1))^{-\frac{2}{3}} \, dx . $$First, simplify the expression inside the integral if possible. Notice that $$ (x^{2} - 1)(x + 1) = (x - 1)(x + 1)(x + 1) = (x - 1)(x + 1)^{2} . So $$the integral becomes $$ \int \left( (x - 1)(x + 1)^{2} ight)^{-\frac{2}{3}} \, dx . $$Apply the substitution $$ u = \arctan x . $$Then, differentiate both sides to find $$ du  in $$terms of $$ dx $$: $$ du = \frac{1}{1 + x^{2}} \, dx $$, which implies $$ dx = (1 + x^{2}) \, du . $$Express the integral entirely in terms of $$ u . $$Since $$ x = 	an u $$, rewrite all parts of the integrand: $$ x - 1 = 	an u - 1 $$, $$ x + 1 = 	an u + 1 $$, and $$ dx = (1 + 	an^{2} u) \, du = \sec^{2} u \, du . $$Substitute these into the integral. Rewrite the integral as $$ \int \left( (	an u - 1)(	an u + 1)^{2} ight)^{-\frac{2}{3}} \sec^{2} u \, du . $$Simplify the expression inside the parentheses if possible, and then simplify the powers and factors to get the integral in terms of $$ u $$ only. Once the integral is expressed in terms of $$ u $$, proceed to integrate with respect to $$ u . $$After finding the antiderivative, substitute back $$ u = \arctan x  to $$express the answer in terms of $$ x .$$

Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
c. u = arctan x
What is the value of the integral?

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Key Concepts

Integration by Substitution

Trigonometric Substitution

Evaluating Integrals with Inverse Trigonometric Functions