Could someone answer this question so I understand it. Thanks
Find the indefinite integral:
�çx/�ã3x^2+4 dx
Use C as the arbitrary constant.
To find the indefinite integral of x/(3x^2 + 4) dx, we can use a substitution to simplify the integrand.
Let's substitute u = 3x^2 + 4.
Differentiating both sides with respect to x, we get du/dx = 6x.
Rearranging the equation, we have dx = du/(6x).
Substituting dx and u into the integral, we get ∫ (x/(3x^2 + 4)) dx = ∫ ((1/(6x))(x/1))(du).
Simplifying, we have (1/6)∫ (du/u).
Now, integrating 1/u with respect to u, we get ln|u| + C, where C is the constant of integration.
Substituting back u = 3x^2 + 4, we have (1/6) ln|3x^2 + 4| + C.
Therefore, the indefinite integral of x/(3x^2 + 4) dx is (1/6) ln|3x^2 + 4| + C, where C is the arbitrary constant.
Sure! I can explain how to find the indefinite integral of the expression �çx/�ã3x^2+4 dx step by step.
To find the indefinite integral, we can use the power rule for integration. According to the power rule, the integral of x^n with respect to x is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
In this case, we need to rewrite the expression �çx/�ã3x^2+4 dx in a form that we can apply the power rule to. Let's first separate the numerator and denominator:
x / (3x^2 + 4) dx
Now, we can see that the expression has the form x^n, where n = 1. We can rewrite x as x^1:
x^1 / (3x^2 + 4) dx
Now, we are ready to apply the power rule. According to the power rule:
The integral of x^n dx = (1/(n+1)) * x^(n+1)
Using n = 1, we can rewrite the expression as:
(1/2) * x^2 / (3x^2 + 4) dx
Now, we have the expression in a form that we can integrate using the power rule. The indefinite integral of (1/2) * x^2 / (3x^2 + 4) dx can be found by applying the formula from the power rule:
(1/2) * Integral of x^2 / (3x^2 + 4) dx
However, this integral is a bit more complicated than a simple power rule integral. To solve it, we need to use a technique called u-substitution.
Let's define u = 3x^2 + 4. We can find du/dx by differentiating both sides with respect to x, which gives us:
du/dx = 6x
Rearranging this equation, we can write dx in terms of du:
dx = (1 / (6x)) * du
Now, we substitute these values back into the integral:
(1/2) * Integral of (x^2 / (3x^2 + 4)) * dx
= (1/2) * Integral of (x^2 / (u)) * (1 / (6x)) * du
Simplifying the expression, we have:
(1/12) * Integral of (x / u) * du
Now, we can notice that x / u = 1 / (u/x). So, we can rewrite the expression as:
(1/12) * Integral of (1 / (u/x)) * du
Now, we can pull out the constant factor (1/12) and simplify the integral:
(1/12) * Integral of (1 / (u/x)) * du
= (1/12) * Integral of (x/u) * du
Now, we have an integral of the form (1/u) * du, which is a simple power rule integral. Applying the power rule again, we get:
= (1/12) * ln|u| + C
Remember that u = 3x^2 + 4, so substituting it back in, we have:
= (1/12) * ln|3x^2 + 4| + C
Therefore, the indefinite integral of �çx/�ã3x^2+4 dx is (1/12) * ln|3x^2 + 4| + C, where C is the constant of integration.