Evaluate the integral. (Use C for the constant of integration.) Integral of (cos theta)(cos^5(sin...


Evaluate the integral. (Use {eq}C {/eq} for the constant of integration.)

{eq}\int (\cos \theta)(\cos^5 (\sin \theta)) \; \mathrm{d} \theta {/eq}

Integration by Substitution:

With the help of integration by substitution, we can convert a difficult integral to an easier integral. This has the effect of changing the variable and the integrand. When dealing with definite integrals, the limits of integration can also change. For example, if we have to integrate the following function:

{eq}\displaystyle{ \int f ( g ( x ) ) g ^ { \prime } ( x ) d x }{/eq}

then we can substitute {eq}\displaystyle{ u=g(x) }{/eq} and can transform the integral into:

{eq}\displaystyle{ \int f ( u ) d u }{/eq}

which look apparently easy as compared to the previous one. We can use the substitution method as many times as required in the same problem.

Answer and Explanation: 1

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We have,

{eq}\displaystyle{ \int (\cos \theta)(\cos^5 (\sin \theta)) \; \mathrm{d} \theta }{/eq}

We can rewrite this as:

{eq}\displaystyle{ \int...

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Learn more about this topic:

How to Solve Integrals Using Substitution


Chapter 13 / Lesson 5

Explore the steps in integration by substitution. Learn the importance of integration with the chain rule and see the u-substitution formula with various examples.

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