Questions: ∫ tan^(1 / 2)(13 x) sec^(2)(13 x) dx =

Transcript text: $\int \tan ^{1 / 2}(13 x) \sec ^{2}(13 x) d x=$

Solution

Solution Steps

To solve the integral $\int \tan^{1/2}(13x) \sec^2(13x) \, dx$, we can use a substitution method. Notice that the derivative of $\tan(13x)$ is $\sec^2(13x) \times 13$. This suggests using the substitution $u = \tan(13x)$, which simplifies the integral significantly. After substitution, the integral becomes a power function of $u$, which can be integrated easily.

Step 1: Identify the Integral and Use Substitution

We are given the integral $\int \tan^{1/2}(13x) \sec^2(13x) \, dx$. To simplify this, we use the substitution $u = \tan(13x)$. The derivative of $\tan(13x)$ is $13 \sec^2(13x)$, so $du = 13 \sec^2(13x) \, dx$. This implies $dx = \frac{1}{13} \frac{du}{\sec^2(13x)}$.

Step 2: Simplify the Integral

Substituting $u = \tan(13x)$ and $dx = \frac{1}{13} \frac{du}{\sec^2(13x)}$ into the integral, we have: \[ \int u^{1/2} \cdot \sec^2(13x) \cdot \frac{1}{13} \frac{du}{\sec^2(13x)} = \frac{1}{13} \int u^{1/2} \, du \]

Step 3: Integrate the Simplified Expression

The integral $\int u^{1/2} \, du$ is a standard power rule integral: \[ \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C \] Thus, the integral becomes: \[ \frac{1}{13} \cdot \frac{2}{3} u^{3/2} = \frac{2}{39} u^{3/2} + C \]

Step 4: Substitute Back to Original Variable

Substitute back $u = \tan(13x)$ to express the result in terms of $x$: \[ \frac{2}{39} (\tan(13x))^{3/2} + C \]