Question

Integration with Tables Help

I have to find the integral of sec^3(7x), integral of tables says its (1/2 sec(u) tan(u) + 1/2 ln|sec (u) + tan(u)| , i replace u with 7x, is this all i do? my homework says its wrong..

Answer 1

\[\large \int\limits \sec^3(7x)dx\]If we let,\[\large \color{orangered}{u=7x}\]Taking the derivative of our substitution, with respect to x, gives us,\[\large \frac{du}{dx}=7 dx\]Which we can write as,\[\large du=7 dx\]Dividing 7 gives us,\[\large \color{royalblue}{\frac{1}{7}du=dx}\]

So we want to use the orange and blue pieces to replace our integral.\[\large \int\limits\limits \sec^3(\color{orangered}{7x})\color{royalblue}{dx} \qquad \rightarrow \qquad \int\limits\limits \sec^3(\color{orangered}{u})\color{royalblue}{\frac{1}{7}du}\]We'll factor out the 1/7th\[\large \frac{1}{7}\int\limits\limits \sec^3(u)du\]

Answer 2

If you use the table from here,
It looks like our answer will be multiplied by \(\dfrac{1}{7}\).

Answer 3

Woops, small typo in the middle step there.
\[\large \frac{du}{dx}=7\]

Answer 4

ah thank you, i was unaware you still had to take the derivative of u.