Integrate $frac3sqrt-5x^2-4sqrt5x+2$
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Integrate $frac3sqrt-5x^2-4sqrt5x+2$
For this question I first factored out the 3, seeing as its a constant
$$3intfrac1sqrt-5x^2-4sqrt5x+2dx$$
I then noticed that this integral has a similar form to the integral of $arcsin(x)$
$$intfrac1sqrta^2-x^2dx=arcsin(fracxa)+C, vert x vert lt a$$
The similarities between these two integrals is clear, but I dont know how to get from $$-5x^2-4sqrt5x+2$$ to $$a^2-x^2$$
Any help or ideas would be highly appreciated!
calculus integration indefinite-integrals
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up vote
3
down vote
favorite
Integrate $frac3sqrt-5x^2-4sqrt5x+2$
For this question I first factored out the 3, seeing as its a constant
$$3intfrac1sqrt-5x^2-4sqrt5x+2dx$$
I then noticed that this integral has a similar form to the integral of $arcsin(x)$
$$intfrac1sqrta^2-x^2dx=arcsin(fracxa)+C, vert x vert lt a$$
The similarities between these two integrals is clear, but I dont know how to get from $$-5x^2-4sqrt5x+2$$ to $$a^2-x^2$$
Any help or ideas would be highly appreciated!
calculus integration indefinite-integrals
$$ -5x^2-4sqrt5xcolorred-4colorred+4+2 = 6-left(5x^2+4sqrt5x+4right) $$
â Hazem Orabi
Aug 29 at 10:19
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Integrate $frac3sqrt-5x^2-4sqrt5x+2$
For this question I first factored out the 3, seeing as its a constant
$$3intfrac1sqrt-5x^2-4sqrt5x+2dx$$
I then noticed that this integral has a similar form to the integral of $arcsin(x)$
$$intfrac1sqrta^2-x^2dx=arcsin(fracxa)+C, vert x vert lt a$$
The similarities between these two integrals is clear, but I dont know how to get from $$-5x^2-4sqrt5x+2$$ to $$a^2-x^2$$
Any help or ideas would be highly appreciated!
calculus integration indefinite-integrals
Integrate $frac3sqrt-5x^2-4sqrt5x+2$
For this question I first factored out the 3, seeing as its a constant
$$3intfrac1sqrt-5x^2-4sqrt5x+2dx$$
I then noticed that this integral has a similar form to the integral of $arcsin(x)$
$$intfrac1sqrta^2-x^2dx=arcsin(fracxa)+C, vert x vert lt a$$
The similarities between these two integrals is clear, but I dont know how to get from $$-5x^2-4sqrt5x+2$$ to $$a^2-x^2$$
Any help or ideas would be highly appreciated!
calculus integration indefinite-integrals
edited Aug 29 at 15:56
tarit goswami
1,137219
1,137219
asked Aug 29 at 10:09
Pablo
33612
33612
$$ -5x^2-4sqrt5xcolorred-4colorred+4+2 = 6-left(5x^2+4sqrt5x+4right) $$
â Hazem Orabi
Aug 29 at 10:19
add a comment |Â
$$ -5x^2-4sqrt5xcolorred-4colorred+4+2 = 6-left(5x^2+4sqrt5x+4right) $$
â Hazem Orabi
Aug 29 at 10:19
$$ -5x^2-4sqrt5xcolorred-4colorred+4+2 = 6-left(5x^2+4sqrt5x+4right) $$
â Hazem Orabi
Aug 29 at 10:19
$$ -5x^2-4sqrt5xcolorred-4colorred+4+2 = 6-left(5x^2+4sqrt5x+4right) $$
â Hazem Orabi
Aug 29 at 10:19
add a comment |Â
2 Answers
2
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A rather messy completing the square:
$-5x^2-4sqrt5x+2$
$-5left(x^2+frac4sqrt5x5right)+2$
$2-5left(x+frac2sqrt55 right)^2+4$
$(sqrt6)^2-left(sqrt5left(x+frac2sqrt55 right)right)^2$
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
add a comment |Â
up vote
0
down vote
Hint: Write
$$-5x^2-4sqrt5x+2=6-(sqrt5x+2)^2$$
and let substitution $sqrt5x+2=sqrt6sintheta$.
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
A rather messy completing the square:
$-5x^2-4sqrt5x+2$
$-5left(x^2+frac4sqrt5x5right)+2$
$2-5left(x+frac2sqrt55 right)^2+4$
$(sqrt6)^2-left(sqrt5left(x+frac2sqrt55 right)right)^2$
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
add a comment |Â
up vote
1
down vote
accepted
A rather messy completing the square:
$-5x^2-4sqrt5x+2$
$-5left(x^2+frac4sqrt5x5right)+2$
$2-5left(x+frac2sqrt55 right)^2+4$
$(sqrt6)^2-left(sqrt5left(x+frac2sqrt55 right)right)^2$
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
A rather messy completing the square:
$-5x^2-4sqrt5x+2$
$-5left(x^2+frac4sqrt5x5right)+2$
$2-5left(x+frac2sqrt55 right)^2+4$
$(sqrt6)^2-left(sqrt5left(x+frac2sqrt55 right)right)^2$
A rather messy completing the square:
$-5x^2-4sqrt5x+2$
$-5left(x^2+frac4sqrt5x5right)+2$
$2-5left(x+frac2sqrt55 right)^2+4$
$(sqrt6)^2-left(sqrt5left(x+frac2sqrt55 right)right)^2$
edited Aug 29 at 10:25
answered Aug 29 at 10:19
Bruce
526113
526113
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
add a comment |Â
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
that is an incredibly messy completing teh square, youre right. any "easier" methods? or is this the only one?
â Pablo
Aug 29 at 12:44
add a comment |Â
up vote
0
down vote
Hint: Write
$$-5x^2-4sqrt5x+2=6-(sqrt5x+2)^2$$
and let substitution $sqrt5x+2=sqrt6sintheta$.
add a comment |Â
up vote
0
down vote
Hint: Write
$$-5x^2-4sqrt5x+2=6-(sqrt5x+2)^2$$
and let substitution $sqrt5x+2=sqrt6sintheta$.
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Hint: Write
$$-5x^2-4sqrt5x+2=6-(sqrt5x+2)^2$$
and let substitution $sqrt5x+2=sqrt6sintheta$.
Hint: Write
$$-5x^2-4sqrt5x+2=6-(sqrt5x+2)^2$$
and let substitution $sqrt5x+2=sqrt6sintheta$.
answered Aug 29 at 10:15
Nosrati
22k51747
22k51747
add a comment |Â
add a comment |Â
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$$ -5x^2-4sqrt5xcolorred-4colorred+4+2 = 6-left(5x^2+4sqrt5x+4right) $$
â Hazem Orabi
Aug 29 at 10:19