Integrals: functions and by parts

Integrals of trigonometric functions

The integral of sin⁡x\sin x is −cos⁡x+C-\cos x+C, and the integral of cos⁡x\cos x is sin⁡x+C\sin x+C.

∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\begin{gathered} \int \sin x\,dx=-\cos x+C, \\[4pt] \int \cos x\,dx=\sin x+C \end{gathered}

Read the derivative formulas backwards. Since cos⁡\cos differentiates to −sin⁡-\sin, sin⁡\sin integrates to −cos⁡-\cos. Since sin⁡\sin differentiates to cos⁡\cos, cos⁡\cos integrates to sin⁡\sin.

∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\begin{gathered} \int \sin x\,dx=-\cos x+C, \\[4pt] \int \cos x\,dx=\sin x+C \end{gathered}

The sign is the whole game: check by differentiating your answer.

Examples

  1. 01Find the value∫0π/3cos⁡x dx\displaystyle \int_0^{\pi/3} \cos x \, dxAnswerHide
    Answer32\displaystyle \frac{\sqrt{3}}{2}
  2. 02Integrate∫sin⁡x dx\displaystyle \int \sin x \, dxAnswerHide
    Answer−cos⁡x+C\displaystyle -\cos x + C
  3. 03Find the value∫0π/4cos⁡x dx\displaystyle \int_0^{\pi/4} \cos x \, dxAnswerHide
    Answer22\displaystyle \frac{\sqrt{2}}{2}
  4. 04Integrate∫cos⁡x dx\displaystyle \int \cos x \, dxAnswerHide
    Answersin⁡x+C\displaystyle \sin x + C
  5. 05What is this?∫2cos⁡x dx\displaystyle \int 2\cos x\,dxAnswerHide
    Answer2sin⁡x+C\displaystyle 2\sin x+C
  6. 06Find the value∫0π/6cos⁡x dx\displaystyle \int_0^{\pi/6} \cos x \, dxAnswerHide
    Answer12\displaystyle \frac{1}{2}

FAQ

Q1Why does ∫sin⁡x dx\int \sin x\,dx have a minus sign?Show answerHide
A
Because differentiating −cos⁡x-\cos x gives back sin⁡x\sin x. Differentiating your answer to check it prevents sign mistakes.

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Published 2026-09-03 · Updated 2026-09-05