AP Calc AB Anti-Derivative Rules

All of the Anti-Derivative Rules you need to memorize!

Power Rule

The Power Rule states that the antiderivative of x^n is (x^(n+1))/(n+1) + C, where n is a real number and n ≠ -1.

Constant Multiple Rule

The Constant Rule states that the antiderivative of a constant k is kx + C, where C is the constant of integration.

Constant Rule

Multiplier Rule

The Multiplier Rule states that the antiderivative of kf(x) is kF(x) + C, where F(x) is the antiderivative of f(x) and k is a constant.

Addition of Equations Rule

The Addition of Equations Rule states that the antiderivative of the sum of two functions f(x) and g(x) is the sum of their antiderivatives, expressed as F(x) + G(x) + C, where F(x) and G(x) are the antiderivatives of f(x) and g(x), respectively.

Integral of Sin

The antiderivative of sin(x) is -cos(x) + C.

Integral of Tan

The antiderivative of tan(x) is -ln|cos(x)| + C. Remember ABSOLUTE VALUE!

Integral of Secant

The antiderivative of sec(x) is ln|sec(x) + tan(x)| + C. Remember ABSOLUTE VALUE!

Integral of Cosine

The antiderivative of cos(x) is sin(x) + C.

Integral of Cotangent

The antiderivative of cot(x) is ln|sin(x)| + C. Remember ABSOLUTE VALUE!

Integral of Cosecant

The antiderivative of csc(x) is -ln|csc(x) + cot(x)| + C. Remember ABSOLUTE VALUE!

Integral of Secant Squared

The antiderivative of sec²(x) is tan(x) + C.

Integral of Cosecant Squared

The antiderivative of csc²(x) is -cot(x) + C.

Integral of Secant x Tangent

The antiderivative of sec(x)tan(x) is sec(x) + C.

Integral of Cosecant x Cotangent

The antiderivative of csc(x)cot(x) is -csc(x) + C.

Integral of e^x

The antiderivative of e^x is e^x + C.

Integral of 1/x

The antiderivative of 1/x is ln|x| + C. This is because the derivative of ln|x| is 1/x, confirming the relationship. You can’t use the power rule because x^-1 + 1 = 0, an exception from power rule. REMEMBER ABSOLUTE VALUE!

Integral of a^x

The antiderivative of a^x is {a^x}/{ln(a)} + C, where a > 0.