Integrate the following: 2pi sec7x dx,,,, (cos3x)^5 dx?

Question

For the first one, I know pi = 3.14somethingsomething but I'm stuck on everything else. Can you pull that out of the integral sign because it's a constant?

The second one I actually solved!!! Is this right:
(cos3x)^5 = [(cos3x)^6] /6 *times* -sin3x

I have a feeling I'm wrong.

Answer 1

Only one problem per question please.

1) Integrate ∫[2π(sec 7x)] dx.

First let's algebraically manipulate the expression to make it easier to integrate.

2π(sec 7x)
= 2π(sec 7x) {[(sec 7x) + (tan 7x)] / [(sec 7x) + (tan 7x)]}
= 2π {[(sec²7x) + (sec 7x)(tan 7x)] / [(tan 7x) + (sec 7x)]}

Now we can integrate.

∫[2π(sec 7x)] dx

= 2π∫{[(sec²7x) + (sec 7x)(tan 7x)] / [(tan 7x) + (sec 7x)]} dx
Let
u = (tan 7x) + (sec 7x)
du = 7[(sec²7x) + (sec 7x)(tan 7x)] dx
du/7 = [(sec²7x) + (sec 7x)(tan 7x)] dx

= (2π/7)∫(1/u)du = (2π/7)ln(u) + C

= (2π/7)ln|(tan 7x) + (sec 7x)| + C
____________

Only one problem per question please.

I will tell you, however, that your answer to the second question is wrong. Sorry.

Answer 2

right this is one greater way, slightly diverse from the others: ? (a million+e^-a million)/ (a million?e^-x) dx = (a million+e^-a million) ? a million/(a million?e^-x) dx (pull out consistent element) = (a million+e^-a million) ? (a million?e^-x + e^-x)/(a million?e^-x) dx (key step-ruin fundamental into 2 components, one wherein the numerator and denominator cancel and alter right into a relentless, the different wherein the numerator is a relentless circumstances the denominator) = (a million+e^-a million) [ ? a million dx + ? e^-x/(a million?e^-x) dx] permit u = a million?e^-x. Then du = e^-x dx and the fundamental turns into (a million+e^-a million) [ ? a million dx + ? a million/u du] = (a million+e^-a million) (x + ln |u| + C) (carry out surely integration) = (a million+e^-a million) (x + ln |a million?e^-x| + C) (back-substitute)