Chapter 5

If \(v(x)=d f / d x,\) what constants \(C\) make \(1-10\) true? $$ \int_{2}^{b} v(x) d x=f(b)+C $$

Find the derivatives of the following functions \(F(x)\). $$ \int_{1}^{x} \cos ^{2} t d t $$

Find the indefinite integrels in \(1-20\). \(\int \sqrt{2+x} d x \quad(a d d+C)\)

Compute the numbers \(\sum_{n=1}^{4} 1 / n\) and \(\sum_{t=2}^{3}(2 i-3)\).

Find the derivatives of the following functions \(F(x)\). $$ \int_{x}^{1} \cos 3 t d t $$

Find the indefinite integrels in \(1-20\). \(\int \sqrt{3-x} d x\) (always \(\left.+C\right)\)

Compute \(\sum_{j=0}^{3}\left(j^{2}-j\right)\) and \(\sum_{j=1}^{n} 1 / 2^{j}\)

Compute \(R_{n}\) and \(L_{x}\) for \(\int_{0}^{1} x^{3} d x\) and \(n=1,2,10 .\) Either verify (with computer) or use (without computer) the formula \(1^{3}+2^{3}+\cdots+n^{3}=\frac{1}{4} n^{2}(n+1)^{2}\)

Find an antiderivative \(f(x)\) for \(v(x)\). Then compute the definite integral \(\int_{0}^{1} r(x) d x=f(1)-f(0)\). $$ 1 \sqrt{x}\left(\text { or } x^{-1 / 2}\right) $$

Evaluate the sum \(\sum_{i=0}^{6} 2^{i}\) and \(\sum_{i=0}^{n} 2^{i} .\)

Find the indefinite integrels in \(1-20\). \(\int(x+1)^{-n} d x\)

About sums \(f_{j}\) and differences \(\boldsymbol{v}_{j}\). Any constant \(C\) can be added to the antiderivative \(f(x)\) because the ________ of a constant is zero. Any \(C\) can be added to \(f_{0}, f_{1}, \ldots\) because the _______ between the \(f^{3} \mathrm{~s}\) is not changed.

If \(v(x)=d f / d x,\) what constants \(C\) make \(1-10\) true? $$ \int_{\mathbb{k / 2}}^{b} v(\sin x) \cos x d x=f(\sin b)+C $$

Find the derivatives of the following functions \(F(x)\). $$ \int_{1}^{x^{2}} u^{3} d u $$

Find the indefinite integrels in \(1-20\). \(\int\left(x^{2}+1\right)^{5} x d x\)

About sums \(f_{j}\) and differences \(\boldsymbol{v}_{j}\). Show that \(f_{j}=r_{i}^{j}(r-1)\) has \(f_{j}-f_{j-1}=r^{j}\) '. Therefore the geometric series \(1+r+\cdots+r^{j-1}\) adds up to _____ (remember to subtract \(f_{0}\) ).

Compute \(\pi\) to six places as \(4 \int_{0}^{1} d x /\left(1+x^{2}\right)\), using any rule.

Find the indefinite integrels in \(1-20\). \(\int \sqrt{1-3 x} d x\)

Express these sums in sigma notation: $$ y_{1}-v_{2}+v_{3}-v_{4} \quad v_{1} w_{1}+v_{2} w_{2}+\cdots+v_{n} w_{n} \quad v_{1}+v_{3}+v_{5} $$

Find the derivatives of the following functions \(F(x)\). \(\int_{x}^{x+1} v(t) d t\) (a "running average" of \(v\) )

Find the indefinite integrels in \(1-20\). \(\int \cos ^{3} x \sin x d x\)

Convert these sums to sigma notation: \(a_{0}+a_{1} x+\cdots+a_{n} x^{n} \quad \sin \frac{2 \pi}{n}+\sin \frac{4 \pi}{n}+\cdots+\sin 2 \pi\)

$$ \int_{1}^{3} x d x+\int_{3}^{5} x d x-\int_{5}^{1} x d x= $$

Find the indefinite integrels in \(1-20\). \(\int \cos x d x / \sin ^{3} x\)

The binomial formula uses coefficients \(\left(\begin{array}{l}n \\\ j\end{array}\right)=\frac{n !}{j !(n-j) !}:\) \((a+b)^{n}=\left(\begin{array}{l}n \\ 0\end{array}\right) a^{n}+\left(\begin{array}{l}n \\ 1\end{array}\right) a^{n-1} b+\cdots+\left(\begin{array}{l}n \\ n\end{array}\right) b^{n}=\sum_{j=0}^{n}\)

Are 9-16 true or false? Give a reason or an example. The minimum of \(\int_{4}^{x} v(t) d t\) is at \(x=4\).

Find the derivatives of the following functions \(F(x)\). $$ \frac{1}{x} \int_{0}^{x} \sin ^{2} t d t $$

Find the indefinite integrels in \(1-20\). \(\int \cos ^{3} 2 x \sin 2 x d x\)

With electronic help compute \(\sum_{1}^{100} 1 j\) and \(\sum_{1}^{1000} 1 / j\)

If \(v(x)=d f / d x,\) what constants \(C\) make \(1-10\) true? $$ \int_{0}^{2} v(x) d x=C \int_{0}^{1} v(2 t) d t . $$

Find the derivatives of the following functions \(F(x)\). $$ \frac{1}{2} \int^{x+2} t^{3} d t $$

Find the indefinite integrels in \(1-20\). \(\int \cos ^{3} x \sin 2 x d x\)

Find the derivatives of the following functions \(F(x)\). $$ \int_{0}^{x}\left[\int_{0}^{t} v(u) d u\right] d t $$

Find the indefinite integrels in \(1-20\). \(\int d t / \sqrt{1-t^{2}}\)

Find an antiderivative \(f(x)\) for \(v(x)\). Then compute the definite integral \(\int_{0}^{1} r(x) d x=f(1)-f(0)\). $$ \sin ^{2} x \cos x $$

Find the indefinite integrels in \(1-20\). \(\int t \sqrt{1-t^{2}} d t\)

Show that \(\left(\sum_{i=1}^{n} a_{l}\right)^{2} \neq \sum_{i=1}^{n} a_{i}^{2}\) and \(\sum_{i=1}^{n} a_{i} b_{i} \neq \sum_{j=1}^{n} a_{j} \sum_{k=1}^{n} b_{k}\)

To compute in \(2=\int_{1}^{2} d x / x=.69315\) with error less than .001 , how many intervals should \(T_{n}\) need? Its leading error is \((\Delta x)^{2}\left[y^{\prime}(b)-y^{\prime}(a)\right] / 12\). Test the actual error with \(y=1 / x\)

Find the indefinite integrels in \(1-20\). \(\int t^{3} d t / \sqrt{1+t^{2}}\)

Telescope" the sums \(\sum_{1=1}^{n}\left(2^{k}-2^{k-1}\right)\) and \(\sum_{j=1}^{10}\left(\frac{1}{j+1}-\frac{1}{j}\right)\) All but two terms cancel.

Find the indefinite integrels in \(1-20\). \(\int t^{3} \sqrt{1-t^{2}} d t\)

Simplify the sums \(\sum_{j=1}^{n}\left(f_{j}-f_{j-1}\right)\) and \(\sum_{j=3}^{12}\left(f_{j+1}-f_{j}\right)\).

Find the derivatives of the following functions \(F(x)\). $$ \int_{-x}^{x} \sin t^{2} d t $$

Find the indefinite integrels in \(1-20\). \(\int(1+\sqrt{x}) d x / \sqrt{x}\)

True or false: (a) \(\sum_{j=4}^{8} v_{j}=\sum_{i=2}^{6} v_{i-2}\) (b) \(\sum_{i=1}^{9} v_{i}=\sum_{i=1}^{11} v_{1-2}\)

Are 9-16 true or false? Give a reason or an example. (a) Antiderivatives of even functions are odd functions. (b) Squares of odd functions are odd functions.

Find the derivatives of the following functions \(F(x)\). $$ \int_{-x}^{x} \sin t d t $$

Find the indefinite integrels in \(1-20\). \(\int\left(1+x^{3 / 2}\right) \sqrt{x} d x\)

For \(I=\int_{0}^{1} \sqrt{1-x^{2}} d x,\) the leading error in the trapezoidal rule is \(\quad\) Try \(n=2,4,8\) to defy the prediction.

Find the indefinite integrels in \(1-20\). \(\int \sec x \tan x d x\)