OMG how to solve these questions :O

Hi GerardWay

Yes, Heureka and Alan are VERY impressive !!         

Here are a couple of hints:

3) What is d(sin⁷x)/dx ?

4) (a² - x²)^5/2 → a⁵(1 - (x/a)²)^5/2 Let sin y = x/a

arg(z) and |z| are the same thing, namely the absolute value of the complex number. If 

z = a + ib then arg(z) = |z| = sqrt(a² + b²)

1) Defining cosh z as , express in the form a + bi

(a) $\cosh(5i)$

(b) $\cosh(2+5i)$

Formula:

$\begin{array}{|rcll|} \hline \cosh(a+i\cdot b) &=& \cosh(a) \cos(b) + i\cdot \sinh(a) \sin(b) \\ \hline \end{array}$

(a)

$\begin{array}{|rcll|} \hline \cosh(5i) \qquad a=0 \qquad b = 5 \\ \cosh(5i) &=& \underbrace{\cosh(0)}_{=1} \cos(5\cdot rad) + i\cdot \underbrace{\sinh(0)}_{=0} \sin(5\cdot rad) \\ \cosh(5i) &=& \cos(5\cdot rad) \\ \cosh(5i) &=& 0.28366218546\dots \\ \hline \end{array}$

(b)

$\begin{array}{|rcll|} \hline \cosh(2+5i) \qquad a=2 \qquad b = 5 \\ \cosh(2+5i) &=& \cosh(2)\cos(5\cdot rad) + i \cdot \sinh(2) \sin(5\cdot rad) \\ \cosh(2+5i) &=& 3.76219569108363145956\dots ~ \cdot 0.28366218546\dots \\ &+& i \cdot 3.626860407847018767668\dots (-0.95892427466) \\ \cosh(2+5i) &=& 1.067192651873\dots ~ - i \cdot 3.477884485899\dots \\ \hline \end{array}$

2) Express in the form a + bi:
$\ln(3+4i)$

Formula:

$\begin{array}{|rcll|} \hline \ln(a+i\cdot b)=\ln(\sqrt{a^2+b^2}) + i\cdot \arctan(\frac{y}{x}) \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline \ln(3+4i) \qquad a=3 \qquad b = 4 \\ \ln(3+4i) &=& \ln(\sqrt{3^2+4^2}) + i\cdot \arctan(\frac{y}{x}) \\ \ln(3+4i) &=& \ln(5) + i\cdot 0.92729521800\dots \\ \ln(3+4i) &=& 1.609437912434100\dots + i\cdot 0.92729521800161\dots \\ \hline \end{array}$

WOW HOW DO YOU EVEN DO THAT

Here's a way of doing the first part of 4):

The second integral can be done in similar fashion (though there are higher powers to deal with).  You should find the result is $\frac{5\pi}{256}a^8$

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