How would you integrate tan5xdx ?
Thanks in advance.
Can you use u substitution?
try taking the derivative of
ln(cosx)
that should give you a good hint
I know you get -sin x; I know how to do the problem without the 5, but I don't understand how to do the problem since it is tangent of 5x
The derivative of ln(cosx) is actually
-tanx, not -sinx. "ln" means "natural (base e) logarithm of..
For the integral of tan 5x, let u = 5x
tan 5x = u
du = 5 dx
integral of tan(5x) dx
= integral of (1/5) tan u du
= -(1/5)ln cos u
= -(1/5)ln cos(5x)
Yes, you can use the u-substitution method to integrate the function tan(5x)dx. Here's how you can do it:
1. Start by letting u = 5x. Then differentiate both sides of this equation with respect to x to obtain du/dx = 5.
2. Solve the expression for dx in terms of du. Dividing both sides of du/dx = 5 by 5 gives you du/dx = 1/5. Rearranging, you get dx = du/5.
3. Substitute the new variable and differential into the original integral. The integral of tan(5x)dx can be rewritten as integral of tan(u) * (du/5).
4. Now, you have successfully transformed the integral with respect to x into an integral with respect to u. Solve the integral of tan(u) * (du/5).
Integration of tan(u) can be done by using the method of partial fractions or by rewriting it as (sin(u)/cos(u)). In this case, we will rewrite it as (sin(u)/cos(u)).
5. The integral becomes (1/5) * integral of (sin(u)/cos(u)) du.
6. Now, integrating (sin(u)/cos(u)) with respect to u gives you -ln|cos(u)|. But don't forget to multiply by (1/5) since it's outside the integral.
7. Therefore, the final result is -(1/5) * ln|cos(u)| + C, where C is the constant of integration.
8. Finally, substitute back the original variable. Remember that u = 5x, so the answer is -(1/5) * ln|cos(5x)| + C.
Hope this helps!