Question

$\int\sec^{2}(5x-1)dx=$

• A. $\frac{1}{5}\tan(5x-1)+C$
• B. $5\tan(5x-1)+C$
• C. $\tan(5x-1)+C$
• D. $\cot(5x-1)+C$
• E. $\frac{1}{5}\cot(5x-1)+C$

Hint:

Integration Tip: For any standard integration formula $\int f(x) dx = F(x) + C$, a linear inner argument always integrates exactly to $\int f(ax+b) dx = \frac{1}{a} F(ax+b) + C$. You can bypass the formal u-substitution completely!

Answer 1

The Correct Option is A

Concept:
The integral of $\sec^2(x)$ is a standard elementary form: $\int \sec^2(x) dx = \tan(x) + C$. When the argument is a linear expression like $(ax + b)$, the Chain Rule in reverse dictates that we simply divide the final anti-derivative by the coefficient $a$.

Step 1: Identify the standard integral form.
The given integral involves the secant squared function: $$I = \int \sec^2(5x - 1) dx$$ We know the base anti-derivative of $\sec^2(u)$ is $\tan(u)$.

Step 2: Set up a linear u-substitution.
Let $u$ equal the inner linear expression: $$u = 5x - 1$$

Step 3: Find the differential du.
Differentiate $u$ with respect to $x$: $$\frac{du}{dx} = 5 \implies dx = \frac{du}{5}$$

Step 4: Substitute into the integral.
Replace the expression and $dx$ with $u$ and $du$: $$I = \int \sec^2(u) \cdot \frac{du}{5}$$ Pull the constant out of the integral: $$I = \frac{1}{5} \int \sec^2(u) du$$

Step 5: Evaluate and substitute x back.
Perform the integration: $$I = \frac{1}{5} \tan(u) + C$$ Replace $u$ with $(5x - 1)$: $$I = \frac{1}{5} \tan(5x - 1) + C$$ Hence the correct answer is (A) $\frac{1{5}\tan(5x-1)+C$}.