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Find the indefinite integral: (Use € for the constant of integration ) f ~ecs_ 3) dx...

Question

Find the indefinite integral: (Use € for the constant of integration ) f ~ecs_ 3) dx

Find the indefinite integral: (Use € for the constant of integration ) f ~ecs_ 3) dx



Answers

Find the indefinite integral. $$\int \operatorname{sech}^{2}(3 x) d x$$

Hi there. In this exercise we need to integrate this function X cube. So it's very simple. This corresponds to just taking X to the fourth power And divided by four. Just to remind you when you have this kind of phenomenal functions to take the power of the Of X or the variable. And then you need to Act Plus one. This case four and then you need to divide by the power that is rising. The variables. In this case we divide by one by four And we obtain X four divided for and never for in the final integral. Never forget diplomacy.

Okay, so we're gonna be finding the value of the integral of three times dx. And so whenever we have a constant multiplied by um the function that we are taking, the integral of, we can actually take that constant outside of the integral and multiply it by just the integral of that function. So this is equal to three times the integral of dx. In the integral of dx. It's just gonna be equal to X. So this is gonna be three times X, Which is equal to three X. And then we also need to add our constant of integration. Since if we took the derivative of this function and we had a constant here, that constant would go to zero and so it would disappear when we take the derivative. And so that's why we always have these constants of integration that we need to add when we're finding um indefinite integral. And so the integral here of three D. X is equal to three X plus C.

Even function is X rays to integral of X cubed the X. Now we have to find that indefinite integral. Here it is of the form integral of x rays to the power N. D X, which can be written as one divide by N plus one X, raised to the power N plus one plus C. Now apply this rule to our function, so our function integral of X q d X can be written as x rays to the powerful, divided by four plus C. And this is our final answer.

Given function is integral of three x rays to the power minus two, divide by three D X. Now we have to find the indefinite integral. Here it is of the form integral of C. F. X. D. X. Which can be written us. See integral of S E. X. The X. Now apply this rule, our function integral of three. Exodus to the power minus two, divide by three D X. Can be written as three integral of x rays. To the power minus two, divide by three D X. Now we can see that x rays to the power minus two by three D. X. Is of the form integral of x rays to the power N. D. X. Which can be written as one, divide by N plus one, multiplied by x rays to the power N plus one plus E. Now apply this rule. Our function can be written us three. My replied by one, divide by minus two, divide by three plus one, multiplied by x rays to the power minus two, divide by three plus one plus C. After simplified, we get nine x rays to the power one, divide by three plus C. And this is our final answer.


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