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

Accepted Solution

Answer:The most general anti-derivative of the function is [tex]\frac{3^x}{\ln \left(3\right)}+7\cosh \left(x\right)+C[/tex]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 Fβ€²(x) = f(x), then F(x) is an anti-derivative of f(x).We can use this theoremIf 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 [tex]\int\limits {f(x)} \, dx=F(x)[/tex] means [tex]F'(x) = f(x)[/tex]To find the anti-derivative of a function you need to follow these steps:Apply the sum rule [tex]\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx[/tex][tex]\int \:3^x+7\sinh \left(x\right)dx = \int \:3^xdx+\int \:7\sinh \left(x\right)dx[/tex]The anti-derivative of [tex]3^x[/tex] is [tex]\int \:3^xdx = \frac{3^x}{\ln \left(3\right)}[/tex]Because [tex]\int a^xdx=\frac{a^x}{\ln a}[/tex]The anti-derivative of [tex]7 \cdot sinh(x)[/tex] is [tex]\int \:7\sinh \left(x\right)dx=7\cosh \left(x\right)[/tex]Because [tex]\int \sinh \left(x\right)dx=\cosh \left(x\right)[/tex]So the most general anti-derivative of the function is [tex]\frac{3^x}{\ln \left(3\right)}+7\cosh \left(x\right)+C[/tex]