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:

F(x)=$\frac{3^x}{ln(3)}$+7cosh(x)+C

Step-by-step explanation:

The function is f(x)=3ˣ+7sinh(x), so we can define it as f(x)=g(x)+h(x) where g(x)=3ˣ and h(x)=7sinh(x).

Now we have to find the most general antiderivative of the function this means that we have to calculate $\int\ {f(x)} \, dx$ wich is the same as $\int\ {(g+h)(x)} \, dx$

The sum rule in integration states that the integral of a sum of two functions is equal to the sum of their integrals. Then,

$\int\ {(g+h)(x)} \, dx$ = $\int\ {g(x)} \, dx + \int\ {h(x)} \, dx$

1- $\int\ {g(x)} \, dx =$$\int\ {3^x} \, dx = \frac{3^x}{ln(3)}+C$ this is because of the rule for integration of exponencial functions, this rule is:

$\int\ {a^x} \, dx =\frac{a^x}{ln(x)}$, in this case a=3

2-$\int\ {h(x)} \, dx =$$\int\ {7sinh(x)} \, dx =7\int\ {sinh(x)} \, dx =7cosh(x)+C$ , number seven is a constant (it doesn´t depend of "x") so it "gets out" of the integral.

The result then is:

F(x)= $\int\ {(h+g)(x)} \, dx=\int\ {h(x)} \, dx +\int\ {g(x)} \, dx$

$\int\ {3^x} \, dx +\int\ {7sinh(x)} \, dx = \frac{3^x}{ln(3)} +7cosh(x) + C$

The letter C is added because the integrations is undefined.