Let's solve each of these questions.

Question 17

Integral to solve: $\int x \sqrt{2x + 1} \, dx$

To solve this integral, we can use substitution. Let: $u = 2x + 1 \Rightarrow du = 2 \, dx \Rightarrow dx = \frac{du}{2}$ Since $x = \frac{u - 1}{2}$, substitute $x$ and $dx$ in terms of $u$: $\int x \sqrt{2x + 1} \, dx = \int \frac{u - 1}{2} \sqrt{u} \cdot \frac{du}{2} = \int \frac{u - 1}{4} \cdot u^{1/2} \, du$ Expanding this: $= \int \frac{u^{3/2} - u^{1/2}}{4} \, du = \frac{1}{4} \int \left( u^{3/2} - u^{1/2} \right) \, du$ Now integrate each term: $= \frac{1}{4} \left( \frac{u^{5/2}}{5/2} - \frac{u^{3/2}}{3/2} \right) + C$ Simplifying the constants: $= \frac{1}{4} \left( \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} \right) + C = \frac{1}{10} u^{5/2} - \frac{1}{6} u^{3/2} + C$ Substitute back $u = 2x + 1$: $= \frac{1}{10} (2x + 1)^{5/2} - \frac{1}{6} (2x + 1)^{3/2} + C$

Answer:

$\int x \sqrt{2x + 1} \, dx = \frac{1}{10} (2x + 1)^{5/2} - \frac{1}{6} (2x + 1)^{3/2} + C$

Question 19

Evaluate the definite integrals: $\int_2^4 (x^2 + 3x + 1) \, dx - \int_2^4 (2x^2 - 7x + 1) \, dx$

Evaluate each integral separately:

1. First integral: $\int_2^4 (x^2 + 3x + 1) \, dx$ Find the antiderivative: $= \left[ \frac{x^3}{3} + \frac{3x^2}{2} + x \right]_2^4$ Plug in the limits: $= \left( \frac{4^3}{3} + \frac{3 \cdot 4^2}{2} + 4 \right) - \left( \frac{2^3}{3} + \frac{3 \cdot 2^2}{2} + 2 \right)$ Calculate each term to get the final result for this integral.

2. Second integral: $\int_2^4 (2x^2 - 7x + 1) \, dx$ Find the antiderivative: $= \left[ \frac{2x^3}{3} - \frac{7x^2}{2} + x \right]_2^4$ Plug in the limits: $= \left( \frac{2 \cdot 4^3}{3} - \frac{7 \cdot 4^2}{2} + 4 \right) - \left( \frac{2 \cdot 2^3}{3} - \frac{7 \cdot 2^2}{2} + 2 \right)$ Calculate each term to get the result for this integral.

Finally, subtract the result of the second integral from the first to get the final answer for Question 19.

Would you like detailed calculations for each numerical step or have any questions?

Related Questions

1. How can substitution simplify integration problems?
2. What are common methods for evaluating definite integrals?
3. Why is it important to verify integrals by differentiation?
4. How do you handle integrals involving square roots of linear functions?
5. What are the steps in handling integration with polynomial terms?

Tip

When evaluating definite integrals, simplifying the integrand first (by combining terms if possible) can sometimes make calculations easier.