Evaluate The Integral:${ \int_2^5 (2x-5)(x-3)^9 , Dx }$

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Introduction

In this article, we will delve into the world of calculus and evaluate the given integral. The integral in question is ∫25(2x−5)(x−3)9 dx\int_2^5 (2x-5)(x-3)^9 \, dx. This problem requires a combination of algebraic manipulation and integration techniques to arrive at the final solution. We will break down the problem into manageable steps and provide a clear explanation of each step.

Step 1: Expand the Product

The first step in evaluating the integral is to expand the product (2x−5)(x−3)9(2x-5)(x-3)^9. We can use the distributive property to expand the product.

(2x−5)(x−3)9=(2x−5)(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3)=2x(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3)−5(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3)\begin{aligned} (2x-5)(x-3)^9 &= (2x-5)(x^9-27x^8+486x^7-10935x^6+153090x^5-\\ &\qquad 153090x^4+10935x^3-486x^2+27x-3)\\ &= 2x(x^9-27x^8+486x^7-10935x^6+153090x^5-\\ &\qquad 153090x^4+10935x^3-486x^2+27x-3)-5(x^9-27x^8+486x^7-\\ &\qquad 10935x^6+153090x^5-153090x^4+10935x^3-486x^2+27x-3) \end{aligned}

Step 2: Integrate the Expanded Product

Now that we have expanded the product, we can integrate each term separately. We will use the power rule of integration, which states that ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.

∫(2x(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3)−5(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3)) dx=∫2x(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3) dx−5∫(x9−27x8+486x7−10935x6+153090x5−153090x4+10935x3−486x2+27x−3) dx=2x1111−54x1010+972x99−21870x88+306090x77−306090x66+21870x55−972x44+54x33−3x2−5(x1010−27x99+486x88−10935x77+153090x66−153090x55+10935x44−486x33+27x22−3x)=2x1111−54x1010+972x99−21870x88+306090x77−306090x66+21870x55−972x44+54x33−3x2−x1010+27x99−486x88+10935x77−153090x66+153090x55−10935x44+486x33−27x22+3x\begin{aligned} \int (2x(x^9-27x^8+486x^7-10935x^6+153090x^5-\\ &\qquad 153090x^4+10935x^3-486x^2+27x-3)-5(x^9-27x^8+486x^7-\\ &\qquad 10935x^6+153090x^5-153090x^4+10935x^3-486x^2+27x-3)) \, dx\\ &= \int 2x(x^9-27x^8+486x^7-10935x^6+153090x^5-\\ &\qquad 153090x^4+10935x^3-486x^2+27x-3) \, dx - 5\int (x^9-27x^8+486x^7-\\ &\qquad 10935x^6+153090x^5-153090x^4+10935x^3-486x^2+27x-3) \, dx\\ &= \frac{2x^{11}}{11} - \frac{54x^{10}}{10} + \frac{972x^9}{9} - \frac{21870x^8}{8} + \frac{306090x^7}{7} - \frac{306090x^6}{6} + \frac{21870x^5}{5} - \frac{972x^4}{4} + \frac{54x^3}{3} - 3x^2 - 5\left(\frac{x^{10}}{10} - \frac{27x^9}{9} + \frac{486x^8}{8} - \frac{10935x^7}{7} + \frac{153090x^6}{6} - \frac{153090x^5}{5} + \frac{10935x^4}{4} - \frac{486x^3}{3} + \frac{27x^2}{2} - 3x\right)\\ &= \frac{2x^{11}}{11} - \frac{54x^{10}}{10} + \frac{972x^9}{9} - \frac{21870x^8}{8} + \frac{306090x^7}{7} - \frac{306090x^6}{6} + \frac{21870x^5}{5} - \frac{972x^4}{4} + \frac{54x^3}{3} - 3x^2 - \frac{x^{10}}{10} + \frac{27x^9}{9} - \frac{486x^8}{8} + \frac{10935x^7}{7} - \frac{153090x^6}{6} + \frac{153090x^5}{5} - \frac{10935x^4}{4} + \frac{486x^3}{3} - \frac{27x^2}{2} + 3x \end{aligned}

Step 3: Simplify the Result

Now that we have integrated each term, we can simplify the result by combining like terms.

2x1111−54x1010+972x99−21870x88+306090x77−306090x66+21870x55−972x44+54x33−3x2−x1010+27x99−486x88+10935x77−153090x66+153090x55−10935x44+486x33−27x22+3x=2x1111−55x1010+999x99−22055x88+308115x77−308115x66+22055x55−999x44+540x33−33x22+3x\begin{aligned} \frac{2x^{11}}{11} - \frac{54x^{10}}{10} + \frac{972x^9}{9} - \frac{21870x^8}{8} + \frac{306090x^7}{7} - \frac{306090x^6}{6} + \frac{21870x^5}{5} - \frac{972x^4}{4} + \frac{54x^3}{3} - 3x^2 - \frac{x^{10}}{10} + \frac{27x^9}{9} - \frac{486x^8}{8} + \frac{10935x^7}{7} - \frac{153090x^6}{6} + \frac{153090x^5}{5} - \frac{10935x^4}{4} + \frac{486x^3}{3} - \frac{27x^2}{2} + 3x\\ &= \frac{2x^{11}}{11} - \frac{55x^{10}}{10} + \frac{999x^9}{9} - \frac{22055x^8}{8} + \frac{308115x^7}{7} - \frac{308115x^6}{6} + \frac{22055x^5}{5} - \frac{999x^4}{4} + \frac{540x^3}{3} - \frac{33x^2}{2} + 3x \end{aligned}

Step 4: Evaluate the Integral

Now that we have simplified the result, we can evaluate the integral by plugging in the upper and lower limits of integration.

$\begin{aligned} \left[\frac{2x^{11}}{11} - \frac{55x^{10}}{10} + \frac{999x^9}{9} - \frac{22055x^8}{8} + \frac{308115x^7}{7} - \frac{308115x^6}{6} + \frac{22055x^5}{5} - \frac{999x^4}{4} + \frac{540x^3}{3} - \frac{33x^2}{2} + 3x\right]_2^5\ &= \left(\frac{2(5)^{11}}{11} - \frac{55(5)^{10}}{10} + \frac{999(5)^9}{9} - \frac{22055(5)^8}{8} + \frac{308115(5)^7}{7} - \frac{308115(5)^6}{6} + \frac{22055(5)^5}{5} - \frac{999(5)^4}{4} + \frac{540(5)^3}{3} - \frac{33(5)^2}{2} + 3(5)\right) - \left(\frac{2(2)^{11}}{11} - \frac{55(2)^{10}}{10} + \frac{999(2)^9}{9} - \frac{22055(2)^8}{8} + \frac{308115(2)^7}{7} - \frac{308115(2)^6}{6} + \frac{22055(2)^5}{5} - \frac{999(2)^4}{4} + \frac{540(2)^3}{3} - \frac{33(2)^2}{2} + 3(2)\right)\ &= \frac{2(1953125)}{11} - \frac{55(9765625)}{10} + \frac{999(1953125)}{9} - \frac{22055(390625)}{8} + \frac{308115(78125)}{7} - \frac{308115(15625)}{6} + \frac{22055(3125)}{5} - \frac{

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Frequently Asked Questions

Q: What is the integral in question?

A: The integral in question is ∫25(2x−5)(x−3)9 dx\int_2^5 (2x-5)(x-3)^9 \, dx.

Q: What is the first step in evaluating the integral?

A: The first step in evaluating the integral is to expand the product (2x−5)(x−3)9(2x-5)(x-3)^9.

Q: How do we expand the product?

A: We can use the distributive property to expand the product.

Q: What is the next step in evaluating the integral?

A: The next step in evaluating the integral is to integrate each term separately.

Q: What rule do we use to integrate each term?

A: We use the power rule of integration, which states that ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.

Q: What is the result of integrating each term?

A: The result of integrating each term is a polynomial expression.

Q: How do we simplify the result?

A: We can simplify the result by combining like terms.

Q: What is the final result of evaluating the integral?

A: The final result of evaluating the integral is a polynomial expression.

Q: How do we evaluate the integral?

A: We evaluate the integral by plugging in the upper and lower limits of integration.

Q: What is the value of the integral?

A: The value of the integral is a numerical value.

Common Misconceptions

Q: Is the integral difficult to evaluate?

A: No, the integral is not difficult to evaluate. It requires a combination of algebraic manipulation and integration techniques.

Q: Do we need to use advanced calculus techniques to evaluate the integral?

A: No, we do not need to use advanced calculus techniques to evaluate the integral. The techniques used are basic and can be applied to a wide range of problems.

Q: Is the integral relevant to real-world problems?

A: Yes, the integral is relevant to real-world problems. It can be used to model and solve problems in physics, engineering, and other fields.

Conclusion

Evaluating the integral ∫25(2x−5)(x−3)9 dx\int_2^5 (2x-5)(x-3)^9 \, dx requires a combination of algebraic manipulation and integration techniques. By following the steps outlined in this article, we can arrive at the final result of the integral. The integral is not difficult to evaluate and can be applied to a wide range of problems in physics, engineering, and other fields.

Additional Resources

For additional resources and information on evaluating integrals, please see the following:

Note: The above article is a Q&A article that provides additional information and resources on evaluating the integral. It is not a replacement for the original article, but rather a supplement to it.