## Substitution Rule

Still remember the Chain Rule? Most important rule in differential calculus. It states that the derivative of F(h(x)) is F'(h(x))h'(x). The product of the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Pretty straightforward, right? Many textbooks will try to indoctrinate you with loads of bullshit concerning variables like u and v.

PART OF THE INTEGRAND WILL BE THE DERIVATIVE OF ANOTHER PART.

Identify those two, and problem solved. Following is a list of functions that can be integrated using the Substitution rule. See if you can identify h(x) and h'(x).

1. f(x) = sin(x)*(cos(x))^2.

2. f(x) = x*(x^2+1)^3.

3. f(x) = e^x * (5*e^x)^2.

1. h(x) = cos(x), h'(x) = sin(x).

2. h(x) = x^2+1, h'(x) = x.

3. h(x) = 5*e^x, h'(x) = e^x.

Got them all? Good for you. Remember, it comes with practice. So,

**The substitution rule, or integration by substitution, says that the indefinite integral of F'(h(x))h'(x) is F(h(x)).**Pretty straightforward, right? Many textbooks will try to indoctrinate you with loads of bullshit concerning variables like u and v.

**FORGET IT!**When using the Substitution rule, look in the integrand for TWO things.**h(x)**, and**h'(x)**.PART OF THE INTEGRAND WILL BE THE DERIVATIVE OF ANOTHER PART.

Identify those two, and problem solved. Following is a list of functions that can be integrated using the Substitution rule. See if you can identify h(x) and h'(x).

1. f(x) = sin(x)*(cos(x))^2.

2. f(x) = x*(x^2+1)^3.

3. f(x) = e^x * (5*e^x)^2.

1. h(x) = cos(x), h'(x) = sin(x).

2. h(x) = x^2+1, h'(x) = x.

3. h(x) = 5*e^x, h'(x) = e^x.

Got them all? Good for you. Remember, it comes with practice. So,

**PRACTICE**,**PRACTICE**, AND**PRACTICE**!