text: # Integrals Involving sin(x) and cos(x) with Odd Power
  
  Tutorial to find integrals involving the product of powers of $$\sin(x)$$ and $$\cos(x)$$ with one of the two having an odd power. Examples and exercises with solutions are included.
  
  ## Examples with Detailed solutions
  
  In what follows, C is the constant of integration.
  
  ### Example 1
  
  Evaluate the integral $\int \sin^3(x) \cos^2(x) dx$
  Solution to Example 1:
  The main idea is to rewrite the integral writing the term with the odd power as the product of a term with power 1 and a term with an even power.
  Example: $$\sin^3(x) = \sin^2(x) \sin(x)$$.
  Hence the given integral may be written as follows:
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = \int \sin^2(x) \cos^2(x) \sin(x) dx$$
  We now use the identity $$\sin^2(x) = 1 - \cos^2(x)$$ and rewrite the given integral as follows:
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = \int (1 - \cos^2(x)) \cos^2(x) \sin(x) dx$$
  We now let $$u = \cos(x)$$, hence $$du/dx = -\sin(x)$$ or $$-du = \sin(x)dx$$ and substitute in the given integral to obtain
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = -\int (1 - u^2) u^2 du$$
  Expand and calculate the integral on the right
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = \int (u^4 - u^2) du$$
  $$= \dfrac{1}{5}u^5 - \dfrac{1}{3}u^3 + C$$
  Substitute $$u$$ by $$\cos(x)$$ to obtain
  $\displaystyle \int \sin^3(x) \cos^2(x) dx = \dfrac{1}{5}\cos^5(x) - \dfrac{1}{3}\cos^3(x) + C$
  
  ### Example 2
  
  Evaluate the integral
  $\int \sin^{12}(x) \cos^5(x) dx$
  Solution to Example 2:
  Rewrite $$\cos^5(x)$$ as follows $$\cos^5(x) = \cos^4(x) \cos(x)$$.
  Hence the given integral may be written as follows:
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int \sin^{12}(x) \cos^4(x) \cos(x) dx$$
  We now use the identity $$\cos^2(x) = 1 - \sin^2(x)$$ to rewrite $$\cos^4(x)$$ in terms of power of $$\sin(x)$$ and rewrite the given integral as follows:
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int \sin^{12}(x) (1 - \sin^2(x))^2 \cos(x) dx$$
  We now let $$u = \sin(x)$$, hence $$du/dx = \cos(x)$$ or $$du = \cos(x)dx$$ and substitute in the given integral to obtain
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int u^{12} (1 - u^2)^2 du$$
  Expand and calculate the integral on the right
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int (u^{16} - 2u^{14} + u^{12}) du$$
  $$= \dfrac{1}{17}u^{17} - \dfrac{2}{15}u^{15} + \dfrac{1}{13}u^{13} + C$$
  Substitute $$u$$ by $$\sin(x)$$ to obtain
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \dfrac{1}{17}\sin^{17}(x) - \dfrac{2}{15}\sin^{15}(x) + \dfrac{1}{13}\sin^{13}(x) + C$$
  
  ## Exercises
  
  Evaluate the following integrals.
  1. $$\displaystyle \int \cos^3(x) \sin^2(x) dx$$
  2. $$\displaystyle \int \sin^3(x) \cos^{14}(x) dx$$
  
  ### Answers to Above Exercises
  
  1. $$-\dfrac{1}{5}\sin^5(x) + \dfrac{1}{3}\sin^3(x)$$
  2. $$\dfrac{1}{17}\cos^{17}(x) - \dfrac{1}{15}\cos^{15}(x)$$
  
  ### More references and Links
  
  integrals and their applications in calculus.
explanation: The document is a tutorial on evaluating integrals involving the product of powers of \(\sin(x)\) and \(\cos(x)\) where one of the two has an odd power. It includes detailed examples with solutions and a set of exercises with answers.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Integrals Involving sin(x) and cos(x) with Odd Power
  
  Tutorial to find integrals involving the product of powers of $$\sin(x)$$ and $$\cos(x)$$ with one of the two having an odd power. Examples and exercises with solutions are included.
  
  ## Examples with Detailed solutions
  
  In what follows, C is the constant of integration.
  
  ### Example 1
  
  Evaluate the integral $\int \sin^3(x) \cos^2(x) dx$
  Solution to Example 1:
  The main idea is to rewrite the integral writing the term with the odd power as the product of a term with power 1 and a term with an even power.
  Example: $$\sin^3(x) = \sin^2(x) \sin(x)$$.
  Hence the given integral may be written as follows:
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = \int \sin^2(x) \cos^2(x) \sin(x) dx$$
  We now use the identity $$\sin^2(x) = 1 - \cos^2(x)$$ and rewrite the given integral as follows:
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = \int (1 - \cos^2(x)) \cos^2(x) \sin(x) dx$$
  We now let $$u = \cos(x)$$, hence $$du/dx = -\sin(x)$$ or $$-du = \sin(x)dx$$ and substitute in the given integral to obtain
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = -\int (1 - u^2) u^2 du$$
  Expand and calculate the integral on the right
  $$\displaystyle \int \sin^3(x) \cos^2(x) dx = \int (u^4 - u^2) du$$
  $$= \dfrac{1}{5}u^5 - \dfrac{1}{3}u^3 + C$$
  Substitute $$u$$ by $$\cos(x)$$ to obtain
  $\displaystyle \int \sin^3(x) \cos^2(x) dx = \dfrac{1}{5}\cos^5(x) - \dfrac{1}{3}\cos^3(x) + C$
  
  ### Example 2
  
  Evaluate the integral
  $\int \sin^{12}(x) \cos^5(x) dx$
  Solution to Example 2:
  Rewrite $$\cos^5(x)$$ as follows $$\cos^5(x) = \cos^4(x) \cos(x)$$.
  Hence the given integral may be written as follows:
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int \sin^{12}(x) \cos^4(x) \cos(x) dx$$
  We now use the identity $$\cos^2(x) = 1 - \sin^2(x)$$ to rewrite $$\cos^4(x)$$ in terms of power of $$\sin(x)$$ and rewrite the given integral as follows:
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int \sin^{12}(x) (1 - \sin^2(x))^2 \cos(x) dx$$
  We now let $$u = \sin(x)$$, hence $$du/dx = \cos(x)$$ or $$du = \cos(x)dx$$ and substitute in the given integral to obtain
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int u^{12} (1 - u^2)^2 du$$
  Expand and calculate the integral on the right
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int (u^{16} - 2u^{14} + u^{12}) du$$
  $$= \dfrac{1}{17}u^{17} - \dfrac{2}{15}u^{15} + \dfrac{1}{13}u^{13} + C$$
  Substitute $$u$$ by $$\sin(x)$$ to obtain
  $$\displaystyle \int \sin^{12}(x) \cos^5(x) dx = \dfrac{1}{17}\sin^{17}(x) - \dfrac{2}{15}\sin^{15}(x) + \dfrac{1}{13}\sin^{13}(x) + C$$
  
  ## Exercises
  
  Evaluate the following integrals.
  1. $$\displaystyle \int \cos^3(x) \sin^2(x) dx$$
  2. $$\displaystyle \int \sin^3(x) \cos^{14}(x) dx$$
  
  ### Answers to Above Exercises
  
  1. $$-\dfrac{1}{5}\sin^5(x) + \dfrac{1}{3}\sin^3(x)$$
  2. $$\dfrac{1}{17}\cos^{17}(x) - \dfrac{1}{15}\cos^{15}(x)$$
  
  ### More references and Links
  
  integrals and their applications in calculus.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a tutorial on evaluating integrals involving the product of powers of \(\sin(x)\) and \(\cos(x)\) where one of the two has an odd power. It includes detailed examples with solutions and a set of exercises with answers.
  
  Label: multiple-question-and-answer