What is $\int cos x - sin (\frac{x}{2} - π)$ $int cosx-sin(x/2-pi)$ ?

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Answer 1 · 2016-11-27T18:12:50

$\int (cos (x) - sin (\frac{x}{2} - π)) d x = sin (x) - \frac{1}{2} cos (\frac{x}{2}) + C$ $int (cos(x) - sin(x/2- pi))dx = sin(x) - 1/2cos(x/2) + C$

Given: $\int (cos (x) - sin (\frac{x}{2} - π)) d x$ $int (cos(x) - sin(x/2- pi))dx$

Use the identity, $sin (A - B) = sin (A) cos (B) - cos (A) sin (B)$ $sin(A - B) = sin(A)cos(B) - cos(A)sin(B)$ :

$sin (\frac{x}{2} - π) = sin (\frac{x}{2}) cos (π) - cos (\frac{x}{2}) sin (π)$ $sin(x/2 - pi) = sin(x/2)cos(pi) - cos(x/2)sin(pi)$

$sin (\frac{x}{2} - π) = - sin (\frac{x}{2})$ $sin(x/2 - pi) = -sin(x/2)$

Substitute into the integrand:

$\int (cos (x) - (- sin (\frac{x}{2}))) d x$ $int (cos(x) - (-sin(x/2)))dx$

$\int (cos (x) + sin (\frac{x}{2})) d x = sin (x) - \frac{1}{2} cos (\frac{x}{2}) + C$ $int (cos(x) + sin(x/2))dx = sin(x) - 1/2cos(x/2) + C$

What is int cosx-sin(x/2-pi) ∫cosx−sin(x2−π)int cosx-sin(x/2-pi) ?