Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. int (dx)/(sqrt(x)+root(4)(x))

Radarfoto67 2022-09-14 Answered
Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. d x x + x 4
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Answers (1)

Grace Moses
Answered 2022-09-15 Author has 13 answers
d x x + x 4
Let x = u 4 d x = 4 u 3 d u
Apply the substitution
d x x + x 4 = 4 u 3 u 2 + u d u
Simplify
4 u 2 u + 1 d u
By the long division
4 u 2 u + 1 d u = ( 4 u 4 + 4 u + 1 ) d u
Integrate
2 u 2 4 u + 4 ln | u + 1 | + C
Back - substitute x = u 4 u = x 4
2 ( x 4 ) 2 4 x 4 + 4 ln | x 4 + 1 | + C
2 x 1 / 2 4 x 4 + 4 ln | x 4 + 1 | + C
Result:
2 x 1 / 2 4 x 4 + 4 ln | x 4 + 1 | + C

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