Question

Use a computer algebra system to evaluate the integral. Compare the answer with the result using tables. If the answer is not the same, show that they are equivalent.

\(\int {\cos e{c^5}xdx} \)

Short answer

Using any computer algebra system,

\(\int {\cos e{c^5}xdx} = \frac{{\left( { - 3\ln \left( {1\cos ec\left( x \right) + \cot \left( x \right)1} \right) + \cot \left( x \right)\cos ec\left. x \right)\left( {2\cos e{c^2}\left( x \right) + 3} \right)} \right)}}{8}\)

Step-by-step solution

using CAS we get  

\(\int {\cos e{c^5}xdx} = \frac{{\left( { - 3\ln \left( {1\cos ec\left( x \right) + \cot \left( x \right)1} \right) + \cot \left( x \right)\cos ec\left. x \right)\left( {2\cos e{c^2}\left( x \right) + 3} \right)} \right)}}{8}\)

\(\frac{{ - 3\ln \left( {1\cos ec\left( x \right) + \cot \left( x \right)1} \right) + \cot \left( x \right)\cos ec\left( x \right)\left( {2\cos e{c^2}\left( x \right) + 3} \right)}}{8}\)

Calculating using table:

I = \(\int {\cos e{c^5}xdx} = \frac{{\left( { - 3\ln \left( {1\cos ec\left( x \right) + \cot \left( x \right)1} \right) + \cot \left( x \right)\cos ec\left. x \right)\left( {2\cos e{c^2}\left( x \right) + 3} \right)} \right)}}{8}\)=\(\int {\cos e{c^3}x'\cos e{c^{^2}}xdx} \)

We know the formula ,

\(\int {\cos e{c^n}udu = \frac{{ - 1}}{{n - 1}}} \cot u\cos e{c^{n - 2}}u + \frac{{n - 2}}{{n - 1}}\int {\cos e{c^{n - 2}}} udu\)

\(\int {\cos e{c^5}xdx} = \frac{{\left( { - 3\ln \left( {1\cos ec\left( x \right) + \cot \left( x \right)1} \right) + \cot \left( x \right)\cos ec\left. x \right)\left( {2\cos e{c^2}\left( x \right) + 3} \right)} \right)}}{8}\)= \(\frac{{ - 1}}{4}\cot x\cos e{c^3}x + \frac{3}{4}\int {\cos e{c^3}xdx} \)

Using the table of integrals,

=\(\) \(\int {\cos e{c^3}udu = \frac{{ - 1}}{2}} \cos ecu\cot u + \frac{1}{2}\ln \left( {\cos ecu - \cot u} \right) + c\)

= \(\int {\cos e{c^5}xdx = \frac{{ - 1}}{4}} \cot x\cos e{c^3}x\frac{3}{4}\left( {\frac{{ - 1}}{2}\cos ecx + \frac{1}{2}\ln \backslash \cos ecx - \cot + c} \right)\)

= \(\frac{{ - 1}}{4}\cot x\cos e{c^3}\frac{{ - 3}}{8}\ln |\cos ec - \cot c| + c\)

We suspect that there are trigonometric identities that show these three answers are equivalent. Indeed,if we also any CAS to simplify these expressions they produce the same answer.

Hence, both answer are equivalent