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Algebra Baldor
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Solve the integral of logarithmic functions $\int y^3\log \left(y\right)dy$

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Solving: $\int y^3\log \left(y\right)dy$
Solve instead: $\int\left(y^3\log y\right)dx$

Solution

Final answer to the problem

$\frac{4y^{4}\ln\left|y\right|-y^{4}}{16\ln\left|10\right|}+C_0$
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Step-by-step Solution

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

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\int y^3\frac{\ln\left(y\right)}{\ln\left(10\right)}dy$

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$\int y^3\frac{\ln\left(y\right)}{\ln\left(10\right)}dy$

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Learn how to solve problems step by step online. Solve the integral of logarithmic functions int(y^3log(y))dy. Change the logarithm to base e applying the change of base formula for logarithms: \log_b(a)=\frac{\log_x(a)}{\log_x(b)}. Multiplying the fraction by y^3. Take the constant \frac{1}{\ln\left|10\right|} out of the integral. We can solve the integral \int y^3\ln\left(y\right)dy by applying integration by parts method to calculate the integral of the product of two functions, using the following formula.

Final answer to the problem

$\frac{4y^{4}\ln\left|y\right|-y^{4}}{16\ln\left|10\right|}+C_0$

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Function Plot

Plotting: $\frac{4y^{4}\ln\left(y\right)-y^{4}}{16\ln\left(10\right)}+C_0$

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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
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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