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The definite integral
equals
.
Solución
Calculate the integral
.
Evaluate the integral by following steps:
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
- step1: Simplify the expression:
- step2: Evaluate the integral:
- step3: Use the substitution
to transform the integral
- step4: Simplify:
- step5: Use the substitution
to transform the integral
- step6: Simplify:
- step7: Rewrite the expression:
- step8: Use properties of integrals:
- step9: Rewrite the fraction:
- step10: Use properties of integrals:
- step11: Calculate:
- step12: Evaluate the integral:
- step13: Evaluate the integral:
- step14: Substitute back:
- step15: Evaluate the power:
- step16: Simplify the expression:
- step17: Multiply the terms:
- step18: Return the limits:
- step19: Calculate the value:
The definite integral evaluates to .
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Beyond the Answer
To evaluate the definite integral
we can simplify the integrand:
Thus, we rewrite the integral:
Now we can split the integral:
Now we will compute each integral separately.
- For the first integral:
Evaluating from 0 to 1:
- For the second integral:
Evaluating from 0 to 1:
Now putting everything together, we have:
Combining these:
Now, we want to find a common denominator to combine these fractions. The common denominator for 5 and 3 is 15:
Combining all terms:
Thus, the final answer for the definite integral is
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