Reverse Power Rule and Reverse Chain Rule

Front

What is the Reverse Power Rule in integration?

Back

The Reverse Power Rule states that the indefinite integral of x^n (where n ≠ -1) is (x^(n+1))/(n+1) + C.

Front

What is the Reverse Chain Rule in integration?

Back

The Reverse Chain Rule is used to integrate composite functions and states that if u = g(x), then ∫f(g(x))g'(x)dx = ∫f(u)du.

Front

What is the indefinite integral of a constant, such as 7?

Back

The indefinite integral of a constant k is kx + C.

Front

How do you integrate a polynomial function like (1/2)x^3 + 7?

Back

You apply the Reverse Power Rule to each term: ∫((1/2)x^3 + 7)dx = (1/8)x^4 + 7x + C.

Front

What is the result of integrating 3x^2(x^3 + 4)^2?

Back

Using the Reverse Chain Rule, the result is (1/3)(x^3 + 4)^3 + C.

Front

How do you evaluate the indefinite integral of -5/x^3 - 2/x^4?

Back

You integrate each term separately: ∫(-5/x^3 - 2/x^4)dx = (5/2)x^(-2) + (2/3)x^(-3) + C.

Front

What is the integral of cos^5(x)sin(x)?

Back

Using the Reverse Chain Rule, the integral is -1/6 cos^6(x) + C.

Front

What is the significance of the constant C in indefinite integrals?

Back

The constant C represents the family of all antiderivatives of a function, accounting for the fact that integration can yield multiple functions differing by a constant.

Front

How do you apply the Reverse Chain Rule to the integral of sin(x)cos^5(x)?

Back

Let u = cos(x), then du = -sin(x)dx, and the integral becomes -∫u^5 du.

Front

What is the general form of the indefinite integral of x^n?

Back

The general form is (x^(n+1))/(n+1) + C, valid for n ≠ -1.