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Find the most general antiderivative of the funct…

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Problem 19 Easy Difficulty

Find the most general antiderivative of the function.
(Check your answers by differentiation.)

$ f(x) = 2^x + 4\sinh x $

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Video Transcript

suppose you have F of X. Which is equal to to raise to expose for hyperbolic sine of X. I mean here we want to find the most general anti derivative of this function. So boys be led F of X as the anti derivative of the function. Now note that the derivative of to raise two X. This is equal to to raise two X times L n F two. And so if you take the derivative of to race to X over Ellen of two, we get two race to X Times Ln of two. This all over Ellen of two. We have to raise two x. So for to raise two X. Our anti derivative would be to raise two X over L N. F. Two. Now for the four times hyperbolic sine of x. We know that the derivative of the hyperbolic cosine of X. This is equal to hyperbolic sine of X. And so if we take the derivative of the Function four times the hyperbolic co sign of X. This will give us four times the hyperbolic sine of x. And so from here we know that the Anti derivative of four times the hyperbolic sine of x. As equal to four times the hyperbolic co sign off X. And so combining these. Now we have F. Of X. Which is the anti derivative of F of X. This is equal to to race to X Over l. n. f. two Plus we have four times the hyperbolic co sign of X. And because you want the general anti derivative we have to add constant C. Were seized any arbitrary real number. Now to check if we have the correct and derivative, we will take the derivative of F of X. Now, since the derivative of F of X, this is equal to derivative of To raise two x over LNF two Plus four co sign H. X. Policy. This is just we have one over LNF two times to race two X Times LNF two Plus We have four times hyperbolic sine of X plus zero. This gives us to raise two X plus four times the hyperbolic sine of x. And so you have the correct and derivative.

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James Stewart

Calculus: Early Transcendentals

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