Question
Use reduction formulas to evaluate the integral.
Ask by Vega Burgess. in the United States
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral of
where
Solution
To evaluate the integral
we can use the reduction formula for integrals of the form
In our case,
Step 1: Apply the Reduction Formula
Using the reduction formula:
Step 2: Evaluate
Now we need to evaluate
Step 3: Evaluate
The integral
Step 4: Substitute Back
Now we substitute this back into the equation for
This simplifies to:
Step 5: Substitute Back into the Original Integral
Now we substitute this result back into the equation for
Step 6: Simplify
Now we simplify this expression:
Final Result
Thus, the integral evaluates to:
where
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Simplify this solution Extra Insights
To evaluate the integral
Let’s adapt this to our integral:
-
Set
and factor out the constant multiplier from the angle: This gives us: -
Applying the reduction formula:
-
Next, we need
which can also be evaluated using the reduction formula: -
The integral
is straightforward: -
Putting it all together:
Substitute back into the equation for: -
Now substitute
into the : -
Now we combine all parts:
This yields the final result involving multiple hyperbolic trigonometric identities and integration by parts. Finally, simplify and remember to substitute back in
And voilà! You’ve evaluated the integral
