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

Answer:

The most general anti-derivative of the function is $\frac{3^x}{\ln \left(3\right)}+7\cosh \left(x\right)+C$

Step-by-step 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

$\int \:3^xdx = \frac{3^x}{\ln \left(3\right)}$

Because $\int a^xdx=\frac{a^x}{\ln a}$

• 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 $\frac{3^x}{\ln \left(3\right)}+7\cosh \left(x\right)+C$