#### Chapter 8 Integral Calculus

Section 8.4 Final Test

# 8.4.1 Final Test Module 1.8

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##### Exercise 8.4.1
Find an antiderivative for each of the following functions:
1. $\int 3\sqrt{x} dx=$
2. $\int \left(2x-e{}^{x+\pi }\right)dx=$
iExpInputHint

##### Exercise 8.4.2
Calculate the integrals
${\int }_{1}^{e}\frac{1}{2x} dx=$
and ${\int }_{5}^{8}\frac{6}{x-4} dx=$

##### Exercise 8.4.3
Calculate the integrals
${\int }_{0}^{3}x·\sqrt{x+1} dx=$
and ${\int }_{\pi }^{\frac{3\pi }{4}}5\mathrm{sin}\left(4x-3\pi \right) dx=$

##### Exercise 8.4.4
Fill in the boxes.
$2{\int }_{a}^{4}|{x}^{3}| dx={\int }_{-4}^{4}|{x}^{3}| dx\mathrm{ }$
$\mathrm{ }|{\int }_{-4}^{4}{x}^{3} dx|$
for $a=$ .
Fill in the boxes such that the statement is correct. Enter comparators as =, <= or >=.

##### Exercise 8.4.5
Calculate the area ${I}_{A}$ of the region $A$ that is bounded by the graphs of the two functions $f$ and $g$ on $\left[-3;2\right]$ with $f\left(x\right)={x}^{2}$ and $g\left(x\right)=6-x$.
Answer: ${I}_{A}=$

##### Exercise 8.4.6
Let an antiderivative $F$ of the function $f$ and an antiderivative $G$ of the function $g$ be given. Moreover, the function $\mathrm{id}$ with $\mathrm{id}\left(x\right)=x$ is given.
Which of the following statements are always true (provided the corresponding combinations/compositions are possible)?
 True? Statement: $\mathrm{id}·F$ is an antiderivative of $\mathrm{id}·f$ $F\circ G$ is an antiderivative of $f\circ g$ $F-G$ is an antiderivative of $f-g$ $F/G$ is an antiderivative of $f/g$ $F·G$ is an antiderivative of $f·g$ $-20·F$ is an antiderivative of $-20·f$

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