Question

$\int 3x^{2}(x^{3}+1)^{10}dx=$

• A. $\frac{(x^{3}+1)^{11}}{11}+C$
• B. $\frac{(x^{3}+1)^{9}}{9}+C$
• C. $\frac{(x^{3}+1)^{11}}{33}+C$
• D. $\frac{(x^{3}+1)^{11}}{11}+x^{3}+C$
• E. $\frac{(x^{3}+1)^{11}}{10}+C$

Hint:

Integration Tip: Always look for function-derivative pairs. If you see $f(x)$ and its derivative $f^{\prime}(x)$ multiplying it somewhere else in the integral, $u = f(x)$ will almost certainly solve the problem.

Answer 1

The Correct Option is A

Concept:
This integral requires integration by substitution (also known as u-substitution). By letting a variable $u$ equal the inner function, its derivative often matches the remaining terms in the integrand, simplifying the problem into a basic power rule integration.

Step 1: Select the substitution variable u.
Let $u$ be the expression inside the parentheses: $$u = x^3 + 1$$

Step 2: Differentiate u to find du.
Take the derivative of $u$ with respect to $x$: $$\frac{du}{dx} = 3x^2$$ $$du = 3x^2\,dx$$ Notice that $3x^2\,dx$ matches perfectly with the remaining terms in the original integral.

Step 3: Rewrite the integral in terms of u.
Substitute $u$ and $du$ into the original integral: $$\int (x^3 + 1)^{10} \cdot (3x^2\,dx)$$ $$= \int u^{10}\,du$$

Step 4: Integrate using the Power Rule.
Use the power rule for integration $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$: $$= \frac{u^{11}}{11} + C$$

Step 5: Substitute x back into the expression.
Replace $u$ with its original definition $(x^3 + 1)$ to get the final answer: $$= \frac{(x^3 + 1)^{11}}{11} + C$$ Hence the correct answer is (A) $\frac{(x^{3}+1)^{11{11}+C$}.