Download
1 / 30
300 likes | 427 Views
warmup. 1). 2). 3). Group Problem: 15 min. solution. 6.1a: Integration rules for Antiderivatives. Find the antiderivative of f(x) = 5x 4 Work backwards (from finding the derivative) The antiderivative of f(x)=F’(x) is x 5.
E N D
warmup 1) 2) 3)
Find the antiderivative of f(x) = 5x4 Work backwards (from finding the derivative) The antiderivative of f(x)=F’(x) is x5
In the example we just did, we know that F(x) = x2 is not the only function whose derivative is f(x) = 2x G(x) = x2 + 2 has 2x as the derivative H(x) = x2 – 7 has 2x as the derivative For any real number, C, the function F(x)=x2 + C has f(x) as an antiderivative
There is a whole family of functions having 2x as an antiderivative This family differs only by a constant
Since the functions G(x) = x2 F(x) = x2 + 2 H(x) = x2 – 7 differ only by a constant, the slope of the tangent line remains the same The family of antiderivatives can be represented by F(x) + C
Integrals such as are called definite integrals because we can find a definite value for the answer. The constant always cancels when finding a definite integral, so we leave it out!
When finding indefinite integrals, we always include the “plus C”. Integrals such as are called indefinite integrals because we can not find a definite value for the answer.
Finding the Antiderivative Finding the antiderivative is the reverse of finding the derivative. Therefore, the rules for derivatives leads to a rule for antiderivatives Example: So
Rules for Antiderivatives Power Rule: Ex: Ex: You can always check your answers by taking the derivative!
You Do 1. 2.
Rules for Finding Antiderivatives Constant Multiple and Sum/Difference:
Recall Previous learning: If f(x) = ex then f’(x) = ex If f(x) = ax then f’(x) = (ln a)ax If f(x) = ekx then f’(x) = kekx If f(x) = akx then f’(x) = k(ln a)akx This leads to the following formulas:
Indefinite Integral of x-1 Note: if x takes on a negative value, then lnx will be undefined. The absolute value sign keeps that from happening.
