Question

To find the value of `int (10x^9 + 10^x * log 10)/(10^x + x^10) dx`, the proper substitution is ______.

Answer 1

To find the value of `int (10x^9 + 10^x * log 10)/(10^x + x^10) dx`, the proper substitution is 10^x + x¹⁰.

Explanation:

Analysis of the Integral

The integral is:

`int (10x^9 + 10^x * log_e 10)/(10^x + x^10) dx`

Let’s test if the numerator is the derivative of the denominator. We set the denominator as our variable t:

t = 10^x + x¹⁰

Now, let's find the derivative of t with respect to `x (dt/dx)`:

1. The derivative of 10^x is 10^x ln 10 (or 10^x log_e 10).

2. The derivative of x¹⁰ is 10x⁹.

Adding these together, we get:

`dt/dx = 10^x log_e 10 + 10x^9`

dt = (10^x log_e 10 + 10x⁹) dx

Notice that this expression for dt is exactly the numerator of our integral.

The proper substitution to solve this integral is:

t = 10^x + x¹⁰ (or any other variable like u = 10^x + x¹⁰)

With this substitution, the integral simplifies to `int 1/t dt`, which result in log |10^x + x¹⁰| + C.