# How to find the integral of x 3^(x^2)

• intenzxboi
In summary, the problem can be solved using the substitution method by letting u = x^2 and then solving for the integral 1/2 3^u du. There is no need to use integration by parts.
intenzxboi

## Homework Statement

$$\int$$ x 3^(x^2)

## Homework Equations

integration by parts

## The Attempt at a Solution

u= x dv= 3^(x^2)
du = 1 dx v=((3^(x^2))/ln 3 ) 2x + c ??

(x)((3^(x^2))/ln 3 ) 2x - $$\int$$ ((3^(x^2))/ln 3 ) 2x dx

is this right??

intenzxboi said:

## Homework Statement

$$\int$$ x 3^(x^2)

## Homework Equations

integration by parts

## The Attempt at a Solution

u= x dv= 3^(x^2)
du = 1 dx v=((3^(x^2))/ln 3 ) 2x + c ??
No: what do you get when you differentiate your v? 3^(x^2) doesn't have a nice integral.

Rather, you should recognise $$\int x3^{x^2} \, dx$$ is built for a substitution: let $$u = x^2$$.

so u=x^2
du = 2x

1/2 3^u du??

intenzxboi said:
so u=x^2
du = 2x

1/2 3^u du??

do you mean

$$\frac{1}{2}\int 3^u du$$

intenzxboi said:

Unco is correct: use the substitution

intenzxboi said:
Then stop listening to those people!

k gotcha. quick question what's the difference between integrating by parts and u substitution

## 1. How do I find the integral of x 3^(x^2)?

To find the integral of x 3^(x^2), you will need to use the substitution method. Let u = x^2, then du = 2x dx. Rewrite the integral as ∫ 3^(u) * (1/2) du. This can be solved using the power rule for integrals, which is ∫ x^n dx = (1/(n+1)) * x^(n+1) + C. Therefore, the integral of x 3^(x^2) is (1/2) * 3^(x^2) + C.

## 2. Can I use u-substitution to find the integral of x 3^(x^2)?

Yes, u-substitution is the most efficient method for finding the integral of x 3^(x^2). By substituting u = x^2, you can simplify the integral and solve using the power rule for integrals.

## 3. Is there another method besides u-substitution to find the integral of x 3^(x^2)?

Yes, there are other methods such as integration by parts or using the logarithmic rule for integrals. However, u-substitution is the most suitable and efficient method for solving this integral.

## 4. Can I use a calculator to find the integral of x 3^(x^2)?

Yes, you can use a graphing calculator or an online integral calculator to find the integral of x 3^(x^2). However, it is always recommended to understand the steps and concepts behind solving the integral rather than relying solely on a calculator.

## 5. What is the significance of finding the integral of x 3^(x^2)?

Finding the integral of x 3^(x^2) is important in various fields of science and mathematics, such as physics, engineering, and statistics. It allows us to calculate the area under a curve and determine important values such as the average of a function. It also helps in solving differential equations and understanding the behavior of exponential functions.

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