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Find the integral of $\left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right)$

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Solution

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Final answer to the problem

$2b^{\left(2x-2\right)}a^2\ln\left(b\right)-4ab^{\left(2x-2\right)}+\frac{a^{\left(3+2x\right)}}{3+2x}+C_0$
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Step-by-step Solution

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Find the integral

$\int\left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right)da$

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$\int\left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right)da$

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Learn how to solve integral calculus problems step by step online. Find the integral of (a^(x+1)-2b^(x-1))(2b^(x-1)+a^(x+1)). Find the integral. Rewrite the integrand \left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right) in expanded form. Expand the integral \int\left(a^{\left(2x+2\right)}-4b^{\left(2x-2\right)}\right)da into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int-4b^{\left(2x-2\right)}da results in: -4ab^{\left(2x-2\right)}+4\int ab^{\left(2x-2\right)}da\ln\left(b\right).

Final answer to the problem

$2b^{\left(2x-2\right)}a^2\ln\left(b\right)-4ab^{\left(2x-2\right)}+\frac{a^{\left(3+2x\right)}}{3+2x}+C_0$

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^(x+1)+-2b^(x-1))(2b^(x-1)+a^(x+1))da using partial fractionsSolve integral of (a^(x+1)+-2b^(x-1))(2b^(x-1)+a^(x+1))da using basic integralsSolve integral of (a^(x+1)+-2b^(x-1))(2b^(x-1)+a^(x+1))da using u-substitutionSolve integral of (a^(x+1)+-2b^(x-1))(2b^(x-1)+a^(x+1))da using integration by partsSolve integral of (a^(x+1)+-2b^(x-1))(2b^(x-1)+a^(x+1))da using trigonometric substitution

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Plotting: $2b^{\left(2x-2\right)}a^2\ln\left(b\right)-4ab^{\left(2x-2\right)}+\frac{a^{\left(3+2x\right)}}{3+2x}+C_0$

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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: Integral Calculus

Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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