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Find the integral of $\ln\left(\frac{\left(3x+22\right)^2\left(-9+e^{2x}\right)}{\sqrt[4]{\left(6x^2+2\right)^3}}\right)$

Step-by-step Solution

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Solution

Final Answer

$-e^{2x}-9\ln\left(e^{2x}-9\right)+e^{2x}\ln\left(e^{2x}-9\right)-\frac{2}{3}\left(3x+22\right)+\frac{44}{3}\ln\left(3x+22\right)+2x\ln\left(3x+22\right)+\frac{3}{2}x-\frac{\sqrt{3}}{2}\arctan\left(\sqrt{3}x\right)-\frac{3}{4}x\ln\left(6x^2+2\right)+C_1$
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Step-by-step Solution

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Find the integral

$\int\ln\left(\frac{\left(3x+22\right)^2\left(-9+e^{2x}\right)}{\sqrt[4]{\left(6x^2+2\right)^3}}\right)dx$

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$\int\ln\left(\frac{\left(3x+22\right)^2\left(-9+e^{2x}\right)}{\sqrt[4]{\left(6x^2+2\right)^3}}\right)dx$

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Learn how to solve problems step by step online. Find the integral of ln(((3x+22)^2(-9+e^(2x)))/((6x^2+2)^3^1/4)). Find the integral. Simplify \sqrt[4]{\left(6x^2+2\right)^3} using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals 3 and n equals \frac{1}{4}. The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Expand the integral \int\left(\ln\left(\left(3x+22\right)^2\left(-9+e^{2x}\right)\right)-\ln\left(\sqrt[4]{\left(6x^2+2\right)^{3}}\right)\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately.

Final Answer

$-e^{2x}-9\ln\left(e^{2x}-9\right)+e^{2x}\ln\left(e^{2x}-9\right)-\frac{2}{3}\left(3x+22\right)+\frac{44}{3}\ln\left(3x+22\right)+2x\ln\left(3x+22\right)+\frac{3}{2}x-\frac{\sqrt{3}}{2}\arctan\left(\sqrt{3}x\right)-\frac{3}{4}x\ln\left(6x^2+2\right)+C_1$

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 ln((3x+22)^2(-9+e^2x)/((6x^2+2)^3^0.25))dx using basic integralsSolve integral of ln((3x+22)^2(-9+e^2x)/((6x^2+2)^3^0.25))dx using u-substitutionSolve integral of ln((3x+22)^2(-9+e^2x)/((6x^2+2)^3^0.25))dx using integration by parts

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Function Plot

Plotting: $-e^{2x}-9\ln\left(e^{2x}-9\right)+e^{2x}\ln\left(e^{2x}-9\right)-\frac{2}{3}\left(3x+22\right)+\frac{44}{3}\ln\left(3x+22\right)+2x\ln\left(3x+22\right)+\frac{3}{2}x-\frac{\sqrt{3}}{2}\arctan\left(\sqrt{3}x\right)-\frac{3}{4}x\ln\left(6x^2+2\right)+C_1$

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(◻)
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◻/◻
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◻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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