Integrals of trigonometric functions

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

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

Examples

1. 01What is this?$\displaystyle \int 2\cos x\,dx$AnswerHide

   Answer$\displaystyle 2\sin x+C$

   Steps

   1. $\cos x$ integrates to $\sin x$, and a constant factor just comes along
   2. $\displaystyle f(x)$$\displaystyle = 2\sin x$$\displaystyle f'(x)$$\displaystyle = 2\cos x$
   3. Differentiating gets us back; no minus appears
2. 02What goes in □?$\displaystyle \int \sin x\,dx = \square\cos x + C$AnswerHide

   Answer$\displaystyle -1$

   Steps

   1. $\cos x$ differentiates to $-\sin x$, but we want $+\sin x$, so put a minus in front
   2. $\displaystyle -(-\sin x)$$\displaystyle = \sin x$
   3. The two minus signs cancel, so $\square$ is $-1$
3. 03What is the value?$\displaystyle \int_{0}^{\frac{\pi}{2}} \sin x\,dx$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\sin x$ integrates to $-\cos x$; subtract the lower end from the upper
   2. $\displaystyle -\cos\frac{\pi}{2}-(-\cos 0)$$\displaystyle = 0+1$$\displaystyle = 1$
4. 04Which one gets a minus sign when integrated?$\displaystyle$A$\displaystyle \sin x$B$\displaystyle \cos x$C$\displaystyle \frac{1}{x}$D$\displaystyle e^{x}$AnswerHide

   Answer$\displaystyle \sin x$

   Steps

   1. Differentiating $-\cos x$ brings back $\sin x$, so integrating $\sin x$ needs the minus
   2. $\displaystyle -(-\sin x)$$\displaystyle = \sin x$
   3. The others do not: $\cos x$ gives $\sin x$, $e^{x}$ gives $e^{x}$, $x^{2}$ gives $\dfrac{x^{3}}{3}$
5. 05Find the value$\displaystyle \int_0^{\pi/3} \cos x \, dx$AnswerHide

   Answer$\displaystyle \frac{\sqrt{3}}{2}$
6. 06Integrate$\displaystyle \int \sin x \, dx$AnswerHide

   Answer$\displaystyle -\cos x + C$

FAQ