#### Chapter 8 Integral Calculus

Section 8.1 Antiderivatives

# 8.1.3 Exercises

##### Exercise 8.1.11
Specify an antiderivative:
1. $\int \left(12{x}^{2}-4{x}^{7}\right)dx$$=$
.
2. $\int \left(\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)\right)dx$$=$
.
3. $\int \frac{1}{6\sqrt{x}} dx$$=$
.

##### Exercise 8.1.12
Find an antiderivative:
1. $\int e{}^{x+2} dx$$=$
.
2. $\int 3x·\sqrt[4]{x} dx$$=$
.
iExpInputHint

##### Exercise 8.1.13
Decide whether the following statements about real-valued functions are true.
 True? Statement: $F$ with $F\left(x\right)=-\frac{\mathrm{cos}\left(\pi x\right)+2}{\pi }$ is an antiderivative of $f$ with $f\left(x\right)=\mathrm{sin}\left(\pi x\right)+2$. $F$ with $F\left(x\right)=-\frac{\mathrm{cos}\left(\pi x\right)+2}{\pi }$ is an antiderivative of $f$ with $f\left(x\right)=\mathrm{sin}\left(\pi x\right)$. $F$ with $F\left(x\right)=-7$ is an antiderivative of $f$ with $f\left(x\right)=-7x$ for $x\in ℝ$. $F$ with $F\left(x\right)=\left(\mathrm{sin}\left(x{\right)\right)}^{2}$ is an antiderivative of $f$ with $f\left(x\right)=2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$. If $F$ is an antiderivative of $f$, and $G$ is an antiderivative of $g$, then $F+G$ is an antiderivative of $f+g$.

##### Exercise 8.1.14
Find an antiderivative of
1. $f\left(x\right):=\frac{8{x}^{3}-6{x}^{2}}{{x}^{4}}$
2. $g\left(x\right):=\frac{18{x}^{2}}{3\sqrt{{x}^{5}}}$
3. $h\left(x\right):=\frac{x+2\sqrt{x}}{4x}$
for $x>0$, after rewriting the terms as reduced sums of fractions:
1. With the simplification $f\left(x\right)$$=$

we have the antiderivative $F\left(x\right)$$=$
for $x>0$.

2. With the simplification $g\left(x\right)$$=$

we have the antiderivative $G\left(x\right)$$=$
for $x>0$.

3. With the simplification $h\left(x\right)$$=$

we have the antiderivative $H\left(x\right)$$=$
for $x>0$.

Enter, for example, $\sqrt{x}$ as sqrt(x).

##### Exercise 8.1.15
Consider a function $f$ with $f\left(x\right)=\frac{1}{x}$ for $x>0$. Moreover, the functions ${F}_{1}$ and ${F}_{2}$ with ${F}_{1}\left(x\right)=\mathrm{ln}\left(7x\right)$ or ${F}_{2}\left(x\right)=\mathrm{ln}\left(x+7\right)$ for $x>0$ are given. Calculate the derivatives of ${F}_{1}$ and ${F}_{2}$, and state whether ${F}_{1}$ and ${F}_{2}$ are antiderivatives of $f$:
We have ${F}_{1}\text{'}\left(x\right)$$=$
and ${F}_{2}\text{'}\left(x\right)$$=$
.
Check the correct answer(s).
${F}_{1}$ is an antiderivative of $f$
${F}_{2}$ is an antiderivative of $f$

##### Exercise 8.1.16
Assume that $F$ is an antiderivative of $f$ with $f\left(x\right)=1+{x}^{2}$, and $F$ has the function value $F\left(0\right)=1$. $F\left(3\right)$ equals
.