question 18 · 1 point\nfind the most general antiderivative (including the constant term, c), f(x) given the…

question 18 · 1 point\nfind the most general antiderivative (including the constant term, c), f(x) given the function below.\nf(x)=-sec²(x)+4csc²(x)\nprovide your answer below:\nf(x)=□

question 18 · 1 point\nfind the most general antiderivative (including the constant term, c), f(x) given the function below.\nf(x)=-sec²(x)+4csc²(x)\nprovide your answer below:\nf(x)=□

Answer

Explanation:

Step1: Recall antiderivative rules

The antiderivative of $\sec^{2}(x)$ is $\tan(x)$ and the antiderivative of $\csc^{2}(x)$ is $-\cot(x)$.

Step2: Find antiderivative of $f(x)$

We have $f(x)=-\sec^{2}(x)+4\csc^{2}(x)$. Using the linear - property of antiderivatives $\int (a\cdot g(x)+b\cdot h(x))dx=a\int g(x)dx + b\int h(x)dx$, where $a=- 1$, $g(x)=\sec^{2}(x)$, $b = 4$, and $h(x)=\csc^{2}(x)$. So $F(x)=-\tan(x)-4\cot(x)+C$.

Answer:

$-\tan(x)-4\cot(x)+C$