Question

The value of $\int (x^2 + 1)^{50} \cdot 2x \, dx$ is given by}

• A. $\frac{(x^2 + 1)^{50}}{50} + C$
• B. $\frac{(x^2 + 1)^{52}}{52} + C$
• C. $\frac{(x^2 + 1)^{51}}{51} + C$
• D. $\frac{(x^2 + 1)^{51}}{50} + C$

Hint:

Whenever the integral has the form $\int [f(x)]^n \cdot f'(x) \, dx$, the direct solution is always $\frac{[f(x)]^{n+1}}{n+1} + C$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To integrate a composite function containing a function and its derivative, we use the method of integration by substitution.
Key Formula or Approach:
The power rule of integration is:
\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \quad (\text{for } n \neq -1) \]

Step 2: Detailed Explanation:
Let us evaluate the integral:
\[ I = \int (x^2 + 1)^{50} \cdot 2x \, dx \]
We choose the substitution:
\[ u = x^2 + 1 \]
Differentiate both sides with respect to $x$:
\[ du = 2x \, dx \]
Substitute $u$ and $du$ into the original integral:
\[ I = \int u^{50} \, du \]
Apply the power rule for integration:
\[ I = \frac{u^{51}}{51} + C \]
Now, substitute back $u = x^2 + 1$:
\[ I = \frac{(x^2 + 1)^{51}}{51} + C \]
Therefore, the value of the integral is $\frac{(x^2 + 1)^{51}}{51} + C$.

Step 3: Final Answer
The correct option is (C).