Evaluate the integral by reversing the order of integration. Integral from 0 to 1 integral from √x to 1 of 5/(y³+1) dydx

⁵/₃ ln 2

Step-by-step explanation:

∫₀¹ ∫√ₓ¹ 5 / (y³ + 1) dy dx

We want to change the order of integration.  To do this, we start by graphing the region contained by the four limits.

x = 0, x = 1

y = √x, y = 1

(Notice that y = √x is the same as x = y²).

Once we've graphed the region, we need to write the domain of x in terms of y.  In this case, 0 < x < y².

Then, we find the range: 0 < y < 1.

Now we can rewrite the integral:

∫₀¹ ∫₀ʸ² 5 / (y³ + 1) dx dy

Notice that the integrand itself, 5 / (y³ + 1), does not change.  Only the limits have changed.

Solve the integral.

∫₀¹ [ 5 / (y³ + 1) x |₀ʸ² ] dy

∫₀¹ [ 5y² / (y³ + 1) ] dy

⁵/₃ ∫₀¹ [ 3y² / (y³ + 1) ] dy

(⁵/₃ ln|y³ + 1|) |₀¹

⁵/₃ ln|1³ + 1| − ⁵/₃ ln|0³ + 1|

⁵/₃ ln 2