Question

$$\int \frac{\tan^4 \sqrt{x} \cdot \sec^2 \sqrt{x}}{\sqrt{x}} \, dx =$$

• A. $-\frac{5}{2} [\tan \sqrt{x}]^5 + c$
• B. $[\tan \sqrt{x}]^5 + c$
• C. $\frac{2}{5} [\tan \sqrt{x}]^5 + c$
• D. $\frac{5}{2} [\tan \sqrt{x}]^5 + c$

Hint:

Recognize compound differentials as a single block to save time. Notice that $d(\tan \sqrt{x}) = \sec^2 \sqrt{x} \cdot \frac{1}{2\sqrt{x}} \, dx$. Therefore, the differential term inside your integrand is simply $2 \cdot d(\tan \sqrt{x})$, meaning the entire problem reduces instantly to $2 \int u^4 \, du$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem requires finding the indefinite integral of a function containing composite trigonometric terms involving $\sqrt{x}$ in both the numerator and the denominator.

Step 2: Key Formula or Approach:
We will use the method of integration by substitution. Since the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$ and the derivative of $\tan\theta$ is $\sec^2\theta$, a multi-stage or chained substitution will cleanly reduce the integrand into a simple power function: $$\int u^n \, du = \frac{u^{n+1}}{n+1} + c$$

Step 3: Detailed Explanation:
Let the given integral be: $$I = \int \frac{\tan^4 \sqrt{x} \cdot \sec^2 \sqrt{x}}{\sqrt{x}} \, dx$$ First, substitute $t = \sqrt{x}$. Differentiating both sides gives: $$dt = \frac{1}{2\sqrt{x}} \, dx \implies \frac{dx}{\sqrt{x}} = 2 \, dt$$ Substitute these expressions back into the integral: $$I = \int \tan^4 t \cdot \sec^2 t \cdot (2 \, dt) = 2 \int \tan^4 t \cdot \sec^2 t \, dt$$ Next, make a second substitution to handle the trigonometric terms. Let $u = \tan t$. Differentiating yields: $$du = \sec^2 t \, dt$$ Substitute these into the integral equation: $$I = 2 \int u^4 \, du$$ Integrate using the standard power rule: $$I = 2 \cdot \left(\frac{u^5}{5}\right) + c = \frac{2}{5} u^5 + c$$ Now, reverse the substitutions systematically. Replace $u$ with $\tan t$: $$I = \frac{2}{5} (\tan t)^5 + c$$ Finally, replace $t$ with $\sqrt{x}$ to get the answer in terms of the original variable: $$I = \frac{2}{5} [\tan \sqrt{x}]^5 + c$$

Step 4: Final Answer:
The value of the integral is $\frac{2}{5} [\tan \sqrt{x}]^5 + c$, which corresponds to option (C).