How Do You Solve an Integral Using Substitution?

In summary, to find the indefinite integral of (square root 2 + lnx) / x, you can use the substitution u = 2 + lnx and then integrate u^(3/2) to get the final answer of (2/3)*(2+lnx)^(3/2) + C.
  • #1
rowdy3
33
0
Use substitution to find each indefinite integral.
∫ (square root 2 + lnx) / x ; dx
I did
u=2+ln(x)
then differentiate both sides to get
du=0+dx/x
∫ (square root 2 + lnx) / x ; dx
∫ (square root u) du
∫ u^0.5 du
u^(3/2)/(3/2)+c
=(2+ln(x))^(3/2) +c
Is the answer right? Thanks.
 
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  • #2
rowdy3 said:
Use substitution to find each indefinite integral.
∫ (square root 2 + lnx) / x ; dx
I did
u=2+ln(x)
then differentiate both sides to get
du=0+dx/x
∫ (square root 2 + lnx) / x ; dx
∫ (square root u) du
∫ u^0.5 du
u^(3/2)/(3/2)+c
=(2+ln(x))^(3/2) +c
Is the answer right? Thanks.

It was until you left out something in the last line.
 
  • #3
(2/3)*[2+lnx]^(3/2)+C. Is that right?
 
Last edited:
  • #4
rowdy3 said:
(2/3)*[2+lnx]^(3/2)+C. Is that right?

Yes, it is.
 

FAQ: How Do You Solve an Integral Using Substitution?

What is integration by substitution?

Integration by substitution is a technique used in calculus to simplify complicated integrals. It involves replacing the variable of integration with a new variable, which allows for easier integration.

How does substitution work in integration?

In substitution, the new variable is chosen in such a way that it cancels out the existing variable and its differential in the integrand. This simplifies the integral, making it easier to evaluate.

When should I use integration by substitution?

Integration by substitution is particularly useful when the integrand involves a composition of functions, such as trigonometric functions or logarithms. It is also helpful when the integrand involves a product of functions, as substitution can help simplify the integral.

What are the steps for integrating by substitution?

The steps for integrating by substitution are as follows: 1) Identify a suitable substitution, 2) Calculate the differential of the new variable, 3) Rewrite the integral in terms of the new variable and its differential, 4) Integrate the new integral, and 5) Substitute back the original variable to obtain the final answer.

Can I always use integration by substitution?

No, integration by substitution can only be used in specific cases where a suitable substitution can be identified. In some cases, other integration techniques such as integration by parts or partial fractions may be more appropriate.

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