Problem of the Week

Updated at Mar 10, 2014 11:58 AM

This week we have another calculus problem:

How would you integrate cos⁡7x\cos{7x}?

Let's start!



∫cos⁡7x dx\int \cos{7x} \, dx

1
Use Integration by Substitution.
Let u=7xu=7x, du=7 dxdu=7 \, dx, then dx=17 dudx=\frac{1}{7} \, du

2
Using uu and dudu above, rewrite ∫cos⁡7x dx\int \cos{7x} \, dx.
∫cos⁡u7 du\int \frac{\cos{u}}{7} \, du

3
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
17∫cos⁡u du\frac{1}{7}\int \cos{u} \, du

4
Use Trigonometric Integration: the integral of cos⁡u\cos{u} is sin⁡u\sin{u}.
sin⁡u7\frac{\sin{u}}{7}

5
Substitute u=7xu=7x back into the original integral.
sin⁡7x7\frac{\sin{7x}}{7}

6
Add constant.
sin⁡7x7+C\frac{\sin{7x}}{7}+C

Done
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