Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)

Accepted Answer

Start by examining the integral $$ \int \frac{\sqrt{x}}{1 - x^{3}} \, dx . $$Notice the hint suggests the substitution $$ u = x^{3/2} . $$Since $$ x^{3/2} = (x^{1/2})^{3} $$, this substitution will help simplify the expression inside the denominator. Express $$ u = x^{3/2} $$ and differentiate both sides with respect to $$ x  to $$find $$ du  in $$terms of $$ dx . $$Recall that $$ \frac{d}{dx} x^{3/2} = \frac{3}{2} x^{1/2} $$, so $$ du = \frac{3}{2} x^{1/2} dx . $$Solve for $$ x^{1/2} dx $$ from the expression for $$ du $$: $$ x^{1/2} dx = \frac{2}{3} du . $$This substitution will allow you to rewrite the integral in terms of $$ u . $$Rewrite the integral in terms of $$ u $$ using the substitutions: $$ \sqrt{x} dx = x^{1/2} dx = \frac{2}{3} du $$ and $$ 1 - x^{3} = 1 - (x^{3/2})^{2} = 1 - u^{2} . $$The integral becomes $$ \int \frac{\frac{2}{3} du}{1 - u^{2}} . $$Recognize that the integral now has the form $$ \int \frac{du}{1 - u^{2}} $$, which is suitable for a trigonometric substitution. Use the substitution $$ u = \sin 	heta  or$$ $$ u = \cos 	heta  to $$simplify the denominator and proceed with the integration.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)

Verified video answer for a similar problem:

Key Concepts

Substitution Method

Trigonometric Substitution

Handling Integrals with Roots and Rational Expressions