Math / Calculus

Question4.) $\int \frac{3 x}{1+9 x^{2}}$

Studdy Solution

STEP 1

1. The integral $\int \frac{3 x}{1+9 x^{2}} \, dx$ is a definite or indefinite integral with respect to $x$ .
2. The integrand $\frac{3 x}{1+9 x^{2}}$ can potentially be simplified using a substitution method.
3. The integral can be approached by recognizing a suitable substitution that simplifies the denominator.

STEP 2

1. Identify a suitable substitution to simplify the integrand.
2. Perform the substitution and simplify the integral.
3. Integrate the simplified form.
4. Substitute back to the original variable.
5. Simplify the result as needed.

STEP 3

Identify a suitable substitution. Let $u = 1 + 9x^2$ , then compute $du$ .
$u = 1 + 9x^2$ $\frac{du}{dx} = 18x$ $du = 18x \, dx$

STEP 4

Solve for $dx$ in terms of $du$ and $x$ .
$du = 18x \, dx \implies dx = \frac{du}{18x}$

STEP 5

Substitute $u$ and $dx$ into the original integral.
$\int \frac{3x}{1+9x^2} \, dx = \int \frac{3x}{u} \cdot \frac{du}{18x}$

STEP 6

Simplify the integrand.
$\int \frac{3x}{u} \cdot \frac{du}{18x} = \int \frac{3}{18} \cdot \frac{1}{u} \, du = \int \frac{1}{6} \cdot \frac{1}{u} \, du$ $= \frac{1}{6} \int \frac{1}{u} \, du$

STEP 7

Integrate $\frac{1}{u}$ with respect to $u$ .
$\frac{1}{6} \int \frac{1}{u} \, du = \frac{1}{6} \ln |u| + C$

STEP 8

Substitute back $u = 1 + 9x^2$ .
$\frac{1}{6} \ln |u| + C = \frac{1}{6} \ln |1 + 9x^2| + C$

STEP 9

Simplify the final result.
The final result is:
$\int \frac{3x}{1+9x^2} \, dx = \frac{1}{6} \ln |1 + 9x^2| + C$

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