Chapter 4: Integration
Section 4.4: Integration by Substitution
Evaluate the definite integral ∫23x3x2+3 ⅆx.
From Example 4.4.4, if u2=x2+3, so that x=2⇒u=7, and x=3⇒u=12, then
=∫712u4−3 u2 ⅆu
From Example 4.4.4, making either the substitution u2=x2+3, or y=x2+3, the indefinite integral of x3x2+3 is x2+35/2/5−x2+33/2. Hence, the value of the definite integral is
Stepwise Maple Solutions
Table 4.4.5(a) contains Maple's annotated stepwise solution obtained with the "Student Calculus1≻All Solution Steps" option in the Context Panel for the inert definite integral.
Tools≻Load Package: Student Calculus 1
∫23x3⁢x2+3 ⅆx→show solution stepsIntegration Steps∫23x3⁢x2+3ⅆx▫1. Apply a change of variables to rewrite the integral in terms of u◦Letx2+3=u2◦Differentiate both sidesⅆⅆxx2+3=ⅆⅆuu2◦Evaluate2⁢x⁢=2⁢u⁢◦Isolate equation for xx=u◦Substitute the values back into the original∫23x3⁢x2+3ⅆx=∫712u4−3⁢u2ⅆuThis gives:∫712u4−3⁢u2ⅆu▫2. Apply the sum rule◦Recall the definition of the sum rule∫f⁡u+g⁡uⅆu=∫f⁡uⅆu+∫g⁡uⅆuf⁡u=u4g⁡u=−3⁢u2This gives:∫712u4ⅆu+∫712−3⁢u2ⅆu▫3. Apply the power rule to the term ∫u4ⅆu◦Recall the definition of the power rule, for n ≠ -1∫uⅆu=◦This means:∫u4ⅆu=◦So,∫u4ⅆu=u55◦Apply limits of definite integral−We can rewrite the integral as:144⁢125−49⁢75+∫712−3⁢u2ⅆu▫4. Apply the constant multiple rule to the term ∫−3⁢u2ⅆu◦Recall the definition of the constant multiple rule∫⁢f⁡uⅆu=⁢∫f⁡uⅆu◦This means:∫−3⁢u2ⅆu=−3⁢∫u2ⅆuWe can rewrite the integral as:144⁢125−49⁢75−3⁢∫712u2ⅆu▫5. Apply the power rule to the term ∫u2ⅆu◦Recall the definition of the power rule, for n ≠ -1∫uⅆu=◦This means:∫u2ⅆu=◦So,∫u2ⅆu=u33◦Apply limits of definite integral−We can rewrite the integral as:84⁢125−14⁢75
Table 4.4.5(a) Maple's annotated stepwise evaluation of the definite integral
A similar solution is available via the
tutor. Pressing the All Steps button and then the Close button will write to the worksheet the solution given in Table 4.4.5(a).
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.
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