Finding Antiderivatives
In Exercises 1–16, find an antid...

Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

4 sec 3x tan 3x

Accepted Answer

Welcome back, everyone. Find the anci derivative of eight second of 5 x multiplied by tangent of 5 x. A 402 5 x plus c. B 5 divided by 8 2nd of 5 x plus c. C negative 8 divided by 5 2nd of 5 x plus c, and D8 divided by 52 of 5 x plus c. For this problem we want to find the antiderivative capital F of X, whose derivative is 825 X. Tangent 5 X. What we can do is simply use the table of derivatives. And recall that the derivative of sequ of a multiplied by x is equal to a multiplied by sequ of ax. Multiplied by a tangent of a x. If we include some constant in front of 2, let's call it c. We can simply multiply both sides by c to show that c multiplied by 2 of a x derivative is going to be equal to c multiplied by a multiplied by 2 of a x multiplied by tangent of a x. And using the properties of derivatives we can just combine c and 2 on the left hand side to show that this expression is the derivative of c. Multiplied by sequence of a x, right, the derivative of that whole function. We know that c multiplied by a multiplied by 2 of a x multiplied by a tangent of a x is given. It is 82 of 5 x tangent of 5 X. So by analogy we can find the coefficient a. Which is the coefficient in front of x and that is equal to 5. If A is equal to 5 and CA. By analogy, is going to be the coefficient in front of the whole function. It is equal to 8. Then we can identify the value of c dividing 8 by a. And that's 8 divided by 5. Well then, so we have c, we have a, and we can now show that the derivative of 8/5. 2nd of 5 eggs. Is equal to the given function. 8th 2nd Of 5 extensions. Of 5 acts. This is one of the antiderivatives to get the whole family. Let's remember that the derivative of a constant is 0. So we just want to add a constant capital C. And this will give us the whole family of an inside derivatives. Now considering the answer choices. We can conclude that the correct answer is option D. Thank you for watching.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.4 sec 3x tan 3x

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Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
4 sec 3x tan 3x