When integrating trigonometric functions containing absolute values, you must identify where the expression inside the absolute value changes sign and split the integral accordingly; for the integral of √(1 - sin(2x)) from 0 to nπ, this simplifies to ∫|cos(x) - sin(x)|dx, which requires splitting at x = π/4 + kπ/2 to handle the sign changes, ultimately yielding the result 2√2n.
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Master the Absolute Value Integral |Integration Skills No.8Hinzugefügt:
Can you solve this tricky integral the squared root of 1 minus sine 2 times X from 0 to n pi?
Today we're tackling integration skill number eight.
First, simplify the integrand.
Using trig identities, the square root becomes the absolute value of cosine X minus sine X. Notice, this function is periodic with period pi.
Now check the the graph. From 0 to pi over 4, cosine is larger. From pi over 4 to pi, sine takes over. So we must split the integral to handle that absolute value. Next, we evaluate. We pull the constant n out and split the bounds. On the first part, integrate cosine X minus sine X. On the second, flip the sign, sine X minus cosine X.
Plug in the limits, simplify, and everything collapses cleanly.
The final result is 2 root 2 times n.
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