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Section 5.7 : Computing Definite Integrals

1. Evaluate each of the following integrals.

  1. \( \displaystyle \int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
  2. \( \displaystyle \int_{{ - 3}}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
  3. \( \displaystyle \int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
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a \( \displaystyle \int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\) Show Solution

This is just an indefinite integral and by this point we should be comfortable doing them so here is the answer to this part.

\[\int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}} = \int{{\cos \left( x \right) - 3{x^{ - 5}}\,dx}} = \sin \left( x \right) + \frac{3}{4}{x^{ - 4}} + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\sin \left( x \right) + \frac{3}{{4{x^4}}} + c}}\]

Don’t forget to add on the “+c” since we are doing an indefinite integral!

b \( \displaystyle \int_{{ - 3}}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\) Show Solution

Recall that in order to do a definite integral the integrand (i.e. the function we are integrating) must be continuous on the interval over which we are integrating, \(\left[ { - 3,4} \right]\) in this case.

We can clearly see that the second term will have division by zero at \(x = 0\) and \(x = 0\) is in the interval over which we are integrating and so this function is not continuous on the interval over which we are integrating.

Therefore, this integral cannot be done.

c \( \displaystyle \int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\) Show Solution

Now, the function still has a division by zero problem in the second term at \(x = 0\). However, unlike the previous part \(x = 0\) does not fall in the interval over which we are integrating, \(\left[ {1,4} \right]\) in this case.

This integral can therefore be done. Here is the work for this integral.

\[\begin{align*}\int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}} & = \int_{1}^{4}{{\cos \left( x \right) - 3{x^{ - 5}}\,dx}} = \left. {\left( {\sin \left( x \right) + \frac{3}{{4{x^4}}}} \right)} \right|_1^4\\ & = \sin \left( 4 \right) + \frac{3}{{4\left( {{4^4}} \right)}} - \left( {\sin \left( 1 \right) + \frac{3}{{4\left( {{1^4}} \right)}}} \right)\\ & = \sin \left( 4 \right) + \frac{3}{{1024}} - \left( {\sin \left( 1 \right) + \frac{3}{4}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\sin \left( 4 \right) - \sin \left( 1 \right) - \frac{{765}}{{1024}}}}\end{align*}\]