Question #71aff

Question

How do you find the integral $\int \frac{1}{x^{2} \cdot \sqrt{x^{2} - 9}} d x$ $int1/(x^2*sqrt(x^2-9))dx$ ?
How do you find the integral $\int \frac{x^{3}}{\sqrt{x^{2} + 9}} d x$ $intx^3/(sqrt(x^2+9))dx$ ?
How do you find the integral $\int x^{3} \cdot \sqrt{9 - x^{2}} d x$ $intx^3*sqrt(9-x^2)dx$ ?
How do you find the integral $\int \frac{x^{3}}{\sqrt{16 - x^{2}}} d x$ $intx^3/(sqrt(16-x^2))dx$ ?
How do you find the integral $\int \frac{\sqrt{x^{2} - 1}}{x} d x$ $intsqrt(x^2-1)/xdx$ ?
How do you find the integral $\int \frac{\sqrt{x^{2} - 9}}{x^{3}} d x$ $intsqrt(x^2-9)/x^3dx$ ?
How do you find the integral $\int \frac{x}{\sqrt{x^{2} + x + 1}} d x$ $intx/(sqrt(x^2+x+1))dx$ ?
How do you find the integral $\int \frac{d t}{\sqrt{t^{2} - 6 t + 13}}$ $intdt/(sqrt(t^2-6t+13))$ ?
How do you find the integral $\int x \cdot \sqrt{1 - x^{4}} d x$ $intx*sqrt(1-x^4)dx$ ?
How do you prove the integral formula $\int \frac{d x}{\sqrt{x^{2} + a^{2}}} = ln (x + \sqrt{x^{2} + a^{2}}) + C$ $intdx/(sqrt(x^2+a^2)) = ln(x+sqrt(x^2+a^2))+ C$ ?

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Answer 1 · 2016-07-12T01:03:48

The answer is the weird decimal number ${cos}^{2} (\sqrt{- 3}) ≅ 0.02577$ $cos^2(sqrt(-3)) ~= 0.02577$ .

The cosine function really only outputs round fractions or whole numbers when some multiple of $π$ $pi$ or a fraction of $π$ $pi$ are input.

For example:
$cos (π) = - 1$ $cos(pi) = -1$
$cos (\frac{π}{2}) = 0$ $cos(pi/2) = 0$
$cos (\frac{π}{4}) = \frac{1}{\sqrt{2}}$ $cos(pi/4) = 1/sqrt(2)$

If you don't have pi in the input, you are guaranteed to receive a decimal output.