Home > Calculus > Integration by Substitution

How do you integrate #3/(4+x^(1/3))#?

Answer 1

William Abdalla

This one is tough!

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Samantha Adkison

To integrate ( \frac{3}{4 + x^{1/3}} ):

Let's use a substitution method:

Let ( u = 4 + x^{1/3} ).

Find ( \frac{du}{dx} ): [ \frac{du}{dx} = \frac{d}{dx}(4 + x^{1/3}) = \frac{1}{3}x^{-2/3} ]

Now, solve for dx: [ dx = 3x^{2/3} du ]

Substitute these into the integral:

[ \int \frac{3}{4 + x^{1/3}} dx = \int \frac{3}{u} \cdot 3x^{2/3} du ] [ = 9 \int \frac{x^{2/3}}{u} du ]

Now, we integrate with respect to u:

[ = 9 \int \frac{x^{2/3}}{u} du ] [ = 9 \int \frac{x^{2/3}}{4 + x^{1/3}} du ]

This integral is not straightforward to solve directly. Typically, a partial fraction decomposition or other advanced integration techniques might be required to solve it completely.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.