Topics

More AI tools from SnapXam

Exercise

$\int\sqrt{x}\ln\left(x\right)dx$

Step-by-step Solution

1

We can solve the integral $\int\sqrt{x}\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

Make diagrams like this with our diagram generator
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sqrt{x}dx}\\ \displaystyle{\int dv=\int \sqrt{x}dx}\end{matrix}$
4

Solve the integral to find $v$

$v=\int\sqrt{x}dx$
5

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $\frac{1}{2}$

$\frac{\sqrt{x^{3}}}{\frac{3}{2}}$
6

Now replace the values of $u$, $du$ and $v$ in the last formula

$\frac{2\sqrt{x^{3}}\ln\left|x\right|}{3}-2\int\frac{\sqrt{x}}{3}dx$
7

The integral $-2\int\frac{\sqrt{x}}{3}dx$ results in: $\frac{-4\sqrt{x^{3}}}{9}$

$\frac{-4\sqrt{x^{3}}}{9}$
8

Gather the results of all integrals

$\frac{2\sqrt{x^{3}}\ln\left|x\right|}{3}+\frac{-4\sqrt{x^{3}}}{9}$
9

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{2\sqrt{x^{3}}\ln\left|x\right|}{3}+\frac{-4\sqrt{x^{3}}}{9}+C_0$

Final answer to the exercise

$\frac{2\sqrt{x^{3}}\ln\left|x\right|}{3}+\frac{-4\sqrt{x^{3}}}{9}+C_0$

Try other ways to solve this exercise

  • Choose an option
  • 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
  • Load more...
Can't find a method? Tell us so we can add it.

Similar Questions Solved

$\int x^3\log\left(x\right)dx$

$\int\ln\left(\sqrt{x}+\sqrt{1+x}\right)dx$

$\int x^2\ln Xdx$

$\int x^2\:lnx\:dx$

$\int\:x\:ln\:2x\:dx$

$\int xLnxdx$

$\int x\:ln\left(2x\right)dx$

Main Topic: Integration by Parts

Integration by parts is an integration method that relates the integral of a product of two or more functions to the integral of their derivative and antiderivative.

Related Topics

Symbolic 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
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

Empowering students and teachers through AI

By signing up, you agree to SnapXam's Terms and Privacy Policy.

Already have an account? Login here
Forgot your password?

Empowering students and teachers through AI



Register here if you don't have an account.
Forgot your password?

Your Personal Math Tutor. Powered by AI

Available 24/7, 365 days a year.

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

Access in depth explanations with descriptive diagrams.

Choose between multiple solving methods.

Download unlimited solutions in PDF format.

Premium access on our iOS and Android apps.

Join 1M+ students worldwide in problem solving.

Choose the plan that suits you best:
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account