What is the intergral of:

2x/[(x+1)^3] dx

Can someone show me how to work to work it out please? Thanks!!

Sure! I'd be happy to show you how to solve this integral step by step.

To find the integral of 2x/[(x+1)^3] dx, we can use a technique called integration by substitution. The idea behind this technique is to substitute a variable with a new one to simplify the integral.

Here's how you can work it out:

Step 1: Start by letting u = x+1. This choice of substitution simplifies the denominator.

Step 2: Calculate du/dx. Since u = x+1, du/dx = 1.

Step 3: Solve the equation from step 1 for x. Subtracting 1 from both sides gives x = u - 1.

Step 4: Substitute the value of x from step 3 and du/dx from step 2 into the integral. This will transform the original integral in terms of the variable u.

The integral then becomes:

∫(2(u-1))/[u^3] du

Step 5: Simplify the numerator of the integrand. Distribute the 2 to get:

∫(2u-2)/[u^3] du

Step 6: Separate the terms in the numerator using the linearity property of integrals:

= ∫(2u/u^3 - 2/u^3) du

Step 7: Split the integral into two separate integrals:

= ∫2u/u^3 du - ∫2/u^3 du

Step 8: Simplify the integrals:

= 2∫du/u^2 - 2∫du/u^3

Step 9: Integrate each term separately. The antiderivative of 1/u^n is -1/(n-1) * u^(1-n), where n is not equal to 1.

This gives us:

= -2/u + 2/(2u^2) + C

Step 10: Substitute back u = x+1:

= -2/(x+1) + 1/(x+1)^2 + C

And that's it! The integral of 2x/[(x+1)^3] dx is equal to -2/(x+1) + 1/(x+1)^2 + C, where C is the constant of integration.