(ax+b),dx=- cos(ax+b)a+C, (ax+b),dx= sin(ax+b)a+C for a 0; angles are radians. The derivatives of cosine and sine are -sin x and cos x. The chain rule adds a factor a which integration must undo. Identify the trigonometric function and coefficient of x. Swap to its antiderivative, divide by that coefficient, retain the negative for sine, and add C. 3sin(2x-1),dx=- 32cos(2x-1)+C; differentiating this gives 3sin(2x-1). Exam check: two negative signs may cancel for a negative inner coefficient. If the argument is constant, integrate the resulting constant instead.

\(\int\sin(ax+b)\,\mathrm{d}x=-\frac{\cos(ax+b)}a+C,\quad\int\cos(ax+b)\,\mathrm{d}x=\frac{\sin(ax+b)}a+C\) for \(a\ne0\); angles are radians.

The derivatives of cosine and sine are \(-\sin x\) and \(\cos x\). The chain rule adds a factor \(a\) which integration must undo.

Identify the trigonometric function and coefficient of \(x\). Swap to its antiderivative, divide by that coefficient, retain the negative for sine, and add \(C\).

\(\int3\sin(2x-1)\,\mathrm{d}x=-\frac32\cos(2x-1)+C\); differentiating this gives \(3\sin(2x-1)\).

Exam check: two negative signs may cancel for a negative inner coefficient. If the argument is constant, integrate the resulting constant instead.

Integrating sine and cosine

Integrating sine and cosine