Chapter 6: Techniques of Integration
Section 6.2: Trigonometric Integrals
Evaluate the indefinite integral ∫sin7 xsin6 x ⅆx.
Using the first formula in Table 6.2.6, namely,
∫sinm xsinn x ⅆx=sinm−n x2m−n−sinm+n x2m+n
with m=7 and n=6, the desired result follows immediately:
∫sin7 xsin6 x ⅆx=sinx2−sin13 x26
Evaluation in Maple
Control-drag the given integral.
Context Panel: Evaluate and Display Inline
∫sin7 xsin6 x ⅆx = 12⁢sin⁡x−126⁢sin⁡13⁢x
Table 6.2.6(a) contains the evaluation of the given integral by the
tutor when the Sum and Constant Multiple rules are taken as Understood Rules.
Table 6.2.6(a) Evaluation of ∫sin7 xsin6 x ⅆx by the Integration Methods tutor
In Table 6.2.6(a), the integration of cosx to sinx is immediate, but the integration of cos13 x must go through the explicit change of variables u=13 x. The most tedious part of the calculation is the entry of the rewrite rule (the first entry in Table 6.2.7).
Table 6.2.6(b) contains a more detailed stepwise solution obtained without any Understood Rules.
Tools≻Load Package: Student Calculus 1
Context Panel: Student Calculus1≻All Solution Steps
∫sin7 xsin6 x ⅆx→show solution stepsIntegration Steps∫sin⁡7⁢x⁢sin⁡6⁢xⅆx▫1. Rewrite◦Equivalent expressionsin⁡7⁢x⁢sin⁡6⁢x=cos⁡x2−cos⁡13⁢x2This gives:∫cos⁡x2−cos⁡13⁢x2ⅆx▫2. Apply the sum rule◦Recall the definition of the sum rule∫f⁡x+g⁡xⅆx=∫f⁡xⅆx+∫g⁡xⅆxf⁡x=cos⁡x2g⁡x=−cos⁡13⁢x2This gives:∫cos⁡x2ⅆx+∫−cos⁡13⁢x2ⅆx▫3. Apply the constant multiple rule to the term ∫cos⁡x2ⅆx◦Recall the definition of the constant multiple rule∫⁢f⁡xⅆx=⁢∫f⁡xⅆx◦This means:∫cos⁡x2ⅆx=∫cos⁡xⅆx2We can rewrite the integral as:∫cos⁡xⅆx2+∫−cos⁡13⁢x2ⅆx▫4. Evaluate the integral of cos(x)◦Recall the definition of the cos rule∫cos⁡xⅆx=sin⁡xThis gives:sin⁡x2+∫−cos⁡13⁢x2ⅆx▫5. Apply the constant multiple rule to the term ∫−cos⁡13⁢x2ⅆx◦Recall the definition of the constant multiple rule∫⁢f⁡xⅆx=⁢∫f⁡xⅆx◦This means:∫−cos⁡13⁢x2ⅆx=−∫cos⁡13⁢xⅆx2We can rewrite the integral as:sin⁡x2−∫cos⁡13⁢xⅆx2▫6. Apply a change of variables to rewrite the integral in terms of u◦Let u beu=13⁢x◦Isolate equation for xx=u13◦Differentiate both sides=13◦Substitute the values for x and dx back into the original∫cos⁡13⁢xⅆx=∫cos⁡u13ⅆuThis gives:sin⁡x2−∫cos⁡u13ⅆu2▫7. Apply the constant multiple rule to the term ∫cos⁡u13ⅆu◦Recall the definition of the constant multiple rule∫⁢f⁡uⅆu=⁢∫f⁡uⅆu◦This means:∫cos⁡u13ⅆu=∫cos⁡uⅆu13We can rewrite the integral as:sin⁡x2−∫cos⁡uⅆu26▫8. Evaluate the integral of cos(u)◦Recall the definition of the cos rule∫cos⁡uⅆu=sin⁡uThis gives:sin⁡x2−sin⁡u26▫9. Revert change of variable◦Variable we defined in step 6u=13⁢xThis gives:sin⁡x2−sin⁡13⁢x26
Table 6.2.6(b) Fully detailed annotated stepwise evaluation of ∫sin7 xsin6 x ⅆx
Note that an annotated stepwise solution is available via the Context Panel with the "All Solution Steps" option.
The rules of integration can also be applied via the Context Panel, as per the figure to the right.
No simple facility is provided for indicating Understood Rules when the stepwise solution is implemented via the Context Panel. Hence, the solution in Table 6.2.6(b) contains the steps where the Sum and Constant Multiple rules are applied, making the solution seem more complicated than it really is.
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