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Integrate the function $\left(\sqrt{a}-\sqrt{x}\right)^2$ from $o$ to $a$

Step-by-step Solution

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Solution

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Final Answer

$-ao+\frac{3}{2}a^2-2\sqrt{a}\left(\frac{2}{3}\sqrt{a^{3}}-\frac{2}{3}\sqrt{o^{3}}\right)-\frac{1}{2}o^2$
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Step-by-step Solution

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Rewrite the integrand $\left(\sqrt{a}-\sqrt{x}\right)^2$ in expanded form

$\int_{o}^{a}\left(a-2\sqrt{a}\sqrt{x}+x\right)dx$

Learn how to solve integrals of rational functions problems step by step online.

$\int_{o}^{a}\left(a-2\sqrt{a}\sqrt{x}+x\right)dx$

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Learn how to solve integrals of rational functions problems step by step online. Integrate the function (a^1/2-x^1/2)^2 from o to a. Rewrite the integrand \left(\sqrt{a}-\sqrt{x}\right)^2 in expanded form. Expand the integral \int_{o}^{a}\left(a-2\sqrt{a}\sqrt{x}+x\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int_{o}^{a} adx results in: a^2-ao. The integral \int_{o}^{a}-2\sqrt{a}\sqrt{x}dx results in: -2\sqrt{a}\left(\frac{2}{3}\sqrt{a^{3}}-\frac{2}{3}\sqrt{o^{3}}\right).

Final Answer

$-ao+\frac{3}{2}a^2-2\sqrt{a}\left(\frac{2}{3}\sqrt{a^{3}}-\frac{2}{3}\sqrt{o^{3}}\right)-\frac{1}{2}o^2$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of ((a^0.5+-1x^0.5)^2)dx from o to a using partial fractionsSolve integral of ((a^0.5+-1x^0.5)^2)dx from o to a using basic integralsSolve integral of ((a^0.5+-1x^0.5)^2)dx from o to a using u-substitutionSolve integral of ((a^0.5+-1x^0.5)^2)dx from o to a using integration by partsSolve integral of ((a^0.5+-1x^0.5)^2)dx from o to a using trigonometric substitution

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Function Plot

Plotting: $\left(\sqrt{a}-\sqrt{x}\right)^2$

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7
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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

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