Integrate the function $\left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right)$

Step-by-step Solution

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Final answer to the problem

$4a^{4}x+4a^{3}x^2+\frac{8}{3}x^{3}a^{2}+8a^{3}x+8a^{2}x-8ax^{4}+4a^2x^2+4ax^2+\frac{x^{5}}{5}+\frac{4}{3}x^{3}a+\frac{4}{3}x^{3}+C_0$
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Step-by-step Solution

How should I solve this problem?

  • Integrate by parts
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • FOIL Method
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1

Find the integral

$\int\left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right)dx$

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$\int\left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right)dx$

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Unlock the first 3 steps of this solution

Learn how to solve problems step by step online. Integrate the function (a^2+(a+x)^2)((a+x)^2+(a+2)^2). Find the integral. Rewrite the integrand \left(a^2+\left(a+x\right)^2\right)\left(\left(a+x\right)^2+\left(a+2\right)^2\right) in expanded form. Expand the integral \int\left(4a^{4}+8a^{3}x+8a^{2}x^{2}+8a^{3}+8a^{2}+4ax^{3}+8a^2x+8ax+x^{4}+4ax^{2}+4x^{2}\right)dx into 11 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int4a^{4}dx results in: 4a^{4}x.

Final answer to the problem

$4a^{4}x+4a^{3}x^2+\frac{8}{3}x^{3}a^{2}+8a^{3}x+8a^{2}x-8ax^{4}+4a^2x^2+4ax^2+\frac{x^{5}}{5}+\frac{4}{3}x^{3}a+\frac{4}{3}x^{3}+C_0$

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Function Plot

Plotting: $4a^{4}x+4a^{3}x^2+\frac{8}{3}x^{3}a^{2}+8a^{3}x+8a^{2}x-8ax^{4}+4a^2x^2+4ax^2+\frac{x^{5}}{5}+\frac{4}{3}x^{3}a+\frac{4}{3}x^{3}+C_0$

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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
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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