Question

Find the most general antiderivative of the function. (Check yo f(x) = 3^x + 7 sinh(x) F(x) = Need Help? Watch It Talk to a Tutor

Answer 1

The most general anti-derivative of the function is
`(3^x)/(\ln \left(3\right))+7\cosh \left(x\right)+C`

Explanation:

Definition. An anti-derivative of a function f(x) is a function whose derivative is equal to f(x). That is, if F′(x) = f(x), then F(x) is an anti-derivative of f(x).

We can use this theorem

If F is an anti-derivative of f on an interval I, then the most general anti-derivative of f on I is

F(x) + C,

where C is an arbitrary constant.

and
`\int\limits {f(x)} \, dx=F(x)` means
`F'(x) = f(x)`

To find the anti-derivative of a function you need to follow these steps:

• Apply the sum rule
  `\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx`

`\int \:3^x+7\sinh \left(x\right)dx = \int \:3^xdx+\int \:7\sinh \left(x\right)dx`

• The anti-derivative of
  `3^x` is

`\int \:3^xdx = (3^x)/(\ln \left(3\right))`

Because
`\int a^xdx=(a^x)/(\ln a)`

• The anti-derivative of
  `7 \cdot sinh(x)` is

`\int \:7\sinh \left(x\right)dx=7\cosh \left(x\right)`

Because
`\int \sinh \left(x\right)dx=\cosh \left(x\right)`

So the most general anti-derivative of the function is
`(3^x)/(\ln \left(3\right))+7\cosh \left(x\right)+C`