Question

Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) f(x) = 7x + 3 sinh(x)

Answer 1

Answer:

$\frac{7}{2}x^{2} +3cosh(x)+C$

Step-by-step explanation:

The antiderivative is found applying the fundamental theorem of calculus:

$\frac{d}{dx}\int\limits^{x}_{a} {f(x)} \, dx=f(x)$

Where a is any number where f(x) is a continuous function.

Here, the antiderivative is $\int\limits^{}_{} {f(x)}dx$

The antiderivative requires to integrate f(x), we apply first the polynomials rule

$\int\limits^{}_{} {x^{n} } \, dx =\frac{1}{n+1}x^{n+1}$

For this case n=1 and the first part of the integral  is:

$\int\limits^{}_{} {7x} \, dx =\frac{7}{2}x^{2}$

The integral of sinh(x) is cosh(x), so, the total antiderivative is

$\frac{7}{2}x^{2}+3cosh(x)+C$

Checking the answer:

$\frac{d(\frac{7}{2}x^{2}+3cosh(x)+C)}{dx}=\frac{7}{2}(2x)+3sinh(x)=7x+3sinh(x)$