Definite integral of piecewise function | AP Calculus AB | Khan Academy

TL;DR

Splitting the definite integral and evaluating each part separately, the overall definite integral is equal to 1/2.

Transcript

• [Voiceover] So we have a f of x right over here and it's defined piecewise for x less than zero, f of x is x plus one, for x is greater than or equal to zero, f of x is cosine of pi x. And we want to evaluate the definite integral from negative one to one of f of x dx. And you might immediately say, well, which of these versions of f of x am I go... Read More

Key Insights

• ❓ Splitting a definite integral can simplify its evaluation by considering the piecewise function separately for each interval.
• ☺️ Evaluating the antiderivative of a function involves incrementing the exponent of x and dividing by the corresponding exponent.
• 😄 The technique of u-substitution can be used to find the antiderivative of certain functions, such as cosine(pi*x).

Summary & Key Takeaways

• The given function, f(x), is defined piecewise as x+1 for x<0 and cosine(pi*x) for x>=0.

• To evaluate the definite integral from -1 to 1 of f(x), the integral is split into two intervals: from -1 to 0 and from 0 to 1.

• The antiderivative of x+1 is evaluated for the first interval, and the antiderivative of cosine(pi*x) is evaluated for the second interval.