👉 Try now NerdPal! Our new math app on iOS and Android

Find the integral $\int\frac{y^3}{y^4+1}dy$

Step-by-step Solution

Go!
Math mode
Text mode
Go!
1
2
3
4
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
Solving: $\int\frac{y^3}{y^4+1}dy$

Final Answer

$\frac{1}{4}\ln\left(1+2\left(y-\frac{\sqrt{2}}{2}\right)^2\right)+\frac{1}{4}\ln\left(1+2\left(y+\frac{\sqrt{2}}{2}\right)^2\right)+C_0$
Got another answer? Verify it here!

Step-by-step Solution

Specify the solving method

1

Rewrite the expression $\frac{y^3}{y^4+1}$ inside the integral in factored form

$\int\frac{y^3}{\left(y^2-\sqrt{2}y+1\right)\left(y^2+\sqrt{2}y+1\right)}dy$

Learn how to solve integrals by partial fraction expansion problems step by step online.

$\int\frac{y^3}{\left(y^2-\sqrt{2}y+1\right)\left(y^2+\sqrt{2}y+1\right)}dy$

Unlock unlimited step-by-step solutions and much more!

Unlock the first 2 steps of this solution.

Learn how to solve integrals by partial fraction expansion problems step by step online. Find the integral int((y^3)/(y^4+1))dy. Rewrite the expression \frac{y^3}{y^4+1} inside the integral in factored form. Rewrite the fraction \frac{y^3}{\left(y^2-\sqrt{2}y+1\right)\left(y^2+\sqrt{2}y+1\right)} in 2 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D. The first step is to multiply both sides of the equation from the previous step by \left(y^2-\sqrt{2}y+1\right)\left(y^2+\sqrt{2}y+1\right). Multiplying polynomials.

Final Answer

$\frac{1}{4}\ln\left(1+2\left(y-\frac{\sqrt{2}}{2}\right)^2\right)+\frac{1}{4}\ln\left(1+2\left(y+\frac{\sqrt{2}}{2}\right)^2\right)+C_0$

Explore different ways to solve this problem

Give us your feedback!

Function Plot

Plotting: $\frac{1}{4}\ln\left(1+2\left(y-\frac{\sqrt{2}}{2}\right)^2\right)+\frac{1}{4}\ln\left(1+2\left(y+\frac{\sqrt{2}}{2}\right)^2\right)+C_0$

SnapXam A2
Answer Assistant

beta
Got a different answer? Verify it!

Go!
1
2
3
4
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

How to improve your answer:

Main Topic: Integrals by Partial Fraction Expansion

The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

Used Formulas

3. See formulas

Your Math & Physics Tutor. Powered by AI

Available 24/7, 365.

Complete step-by-step math solutions. No ads.

Includes multiple solving methods.

Support for more than 100 math topics.

Premium access on our iOS and Android apps as well.

20% discount on online tutoring.

Choose your Subscription plan:
Have a promo code?
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account