∫x√(5x-2) Dx: U-Substitution & Integration By Parts Guide

by Sebastian Müller 58 views

Hey guys! Let's dive into a fun integration problem where we'll tackle the integral ∫x√(5x-2) dx using two powerful methods: U-substitution and integration by parts. Both methods, while different in their approach, will lead us to the same, equivalent answer. Buckle up, it's going to be an exciting ride!

(e) Integrating ∫x√(5x-2) dx

This section will explore how to solve the integral ∫x√(5x-2) dx using both U-substitution and integration by parts. We'll break down each method step-by-step, ensuring you understand the process and can apply it to similar problems. We'll also highlight why both methods give the same answer, emphasizing the beauty and consistency of calculus.

Method 1: U-Substitution

U-substitution, also known as variable substitution, is a technique used to simplify integrals by replacing a complex expression with a single variable, 'u'. The key is to choose a 'u' that, when substituted, makes the integral easier to handle. For our integral, ∫x√(5x-2) dx, a natural choice for 'u' is the expression inside the square root, 5x - 2. Let's see how this works:

  1. Choose u: Let u = 5x - 2.

  2. Find du: Differentiate both sides with respect to x: du/dx = 5, which implies du = 5dx. We can also write dx = du/5.

  3. Express x in terms of u: From u = 5x - 2, we get 5x = u + 2, so x = (u + 2)/5.

  4. Substitute: Now we substitute u, dx, and x in terms of u into the integral:

    ∫x√(5x-2) dx = ∫((u + 2)/5)√u (du/5) = (1/25)∫(u + 2)√u du

  5. Simplify and expand: We can rewrite √u as u^(1/2) and expand the integrand:

    (1/25)∫(u + 2)u^(1/2) du = (1/25)∫(u^(3/2) + 2u^(1/2)) du

  6. Integrate: Now we integrate term by term using the power rule for integration, which states ∫x^n dx = (x^(n+1))/(n+1) + C:

    (1/25)∫(u^(3/2) + 2u^(1/2)) du = (1/25) [(u^(5/2))/(5/2) + 2(u^(3/2))/(3/2)] + C = (1/25) [(2/5)u^(5/2) + (4/3)u^(3/2)] + C

  7. Simplify:

    = (2/125)u^(5/2) + (4/75)u^(3/2) + C

  8. Substitute back: Finally, we substitute u = 5x - 2 back into the expression:

    = (2/125)(5x - 2)^(5/2) + (4/75)(5x - 2)^(3/2) + C

Thus, using U-substitution, we find the integral to be (2/125)(5x - 2)^(5/2) + (4/75)(5x - 2)^(3/2) + C. This looks a bit complex, but we'll see how integration by parts leads to an equivalent answer.

Method 2: Integration by Parts

Integration by parts is another powerful technique used to integrate the product of two functions. It's based on the product rule for differentiation and is particularly useful when dealing with integrals like ours where we have 'x' multiplied by a function of 'x'. The formula for integration by parts is: ∫u dv = uv - ∫v du. The trick lies in choosing appropriate 'u' and 'dv' parts of the integrand.

  1. Choose u and dv: For our integral, ∫x√(5x-2) dx, a good choice is:

    • u = x (because its derivative simplifies things)
    • dv = √(5x-2) dx (the remaining part of the integrand)

  2. Find du and v:

    • Differentiate u with respect to x: du/dx = 1, so du = dx.

    • Integrate dv to find v: v = ∫√(5x-2) dx. To solve this, we can use a simple U-substitution (let w = 5x-2, dw = 5dx, dx = dw/5):

      v = ∫√w (dw/5) = (1/5)∫w^(1/2) dw = (1/5) * (2/3)w^(3/2) = (2/15)(5x-2)^(3/2)

  3. Apply the integration by parts formula: ∫u dv = uv - ∫v du

    ∫x√(5x-2) dx = x * (2/15)(5x-2)^(3/2) - ∫(2/15)(5x-2)^(3/2) dx = (2x/15)(5x-2)^(3/2) - (2/15)∫(5x-2)^(3/2) dx

  4. Integrate the remaining integral: We need to integrate ∫(5x-2)^(3/2) dx. Again, we can use U-substitution (let w = 5x-2, dw = 5dx, dx = dw/5):

    ∫(5x-2)^(3/2) dx = ∫w^(3/2) (dw/5) = (1/5) * (2/5)w^(5/2) = (2/25)(5x-2)^(5/2)

  5. Substitute back and simplify:

    ∫x√(5x-2) dx = (2x/15)(5x-2)^(3/2) - (2/15) * (2/25)(5x-2)^(5/2) + C = (2x/15)(5x-2)^(3/2) - (4/375)(5x-2)^(5/2) + C

So, using integration by parts, we get (2x/15)(5x-2)^(3/2) - (4/375)(5x-2)^(5/2) + C. This looks different from the result we got with U-substitution, but fear not! They are equivalent.

Showing Equivalence

The results from U-substitution and integration by parts look different, but they represent the same integral, so they must be equivalent. Let's rewrite the integration by parts result to show this:

Integration by parts result: (2x/15)(5x-2)^(3/2) - (4/375)(5x-2)^(5/2) + C

Let's find a common denominator and combine the terms:

= (50x/375)(5x-2)^(3/2) - (4/375)(5x-2)^(5/2) + C = (1/375)(5x-2)^(3/2) [50x - 4(5x-2)] + C = (1/375)(5x-2)^(3/2) [50x - 20x + 8] + C = (1/375)(5x-2)^(3/2) [30x + 8] + C = (2/375)(5x-2)^(3/2) [15x + 4] + C

Now, let's expand this:

= (2/375)(5x-2)^(3/2) [15x + 4] + C = (30x/375)(5x-2)^(3/2) + (8/375)(5x-2)^(3/2) + C

U-substitution result: (2/125)(5x - 2)^(5/2) + (4/75)(5x - 2)^(3/2) + C

Let's find a common denominator:

= (6/375)(5x-2)^(5/2) + (20/375)(5x-2)^(3/2) + C

Let's take the first one: (2/125)(5x - 2)^(5/2) = (2/125)(5x-2)(5x-2)^(3/2) = (2/125)(5x-2)(5x-2)^(3/2) = (10x-4)/125 (5x-2)^(3/2) = (30x - 12)/375(5x-2)^(3/2) + C

Let's take the second one: (4/75)(5x - 2)^(3/2) = (20/375)(5x-2)^(3/2) + C

Now, if you find the derivative of both results, the results should be the same.

Let's take the derivative of (2/125)(5x - 2)^(5/2) + (4/75)(5x - 2)^(3/2) + C = (2/125)* 5/2(5)(5x-2)^(3/2) + (4/75)*3/2(5)(5x-2)^(1/2) = (5x-2)^(3/2) / 25 + 2/5(5x-2)^(1/2) = x(5x-2)^(1/2)

Let's take the derivative of (2x/15)(5x-2)^(3/2) - (4/375)(5x-2)^(5/2) + C = (2/15)(5x-2)^(3/2) + (2x/15)(3/2)(5)(5x-2)^(1/2) - (4/375)(5/2)(5)(5x-2)^(3/2) = (2/15)(5x-2)^(3/2) + x(5x-2)^(1/2) - (5x-2)^(3/2)/15 = x(5x-2)^(1/2)

We see both results are equivalent, even though they appear different at first glance. This highlights an important concept in calculus: there can be multiple ways to express the same answer.

(f) Finding the Equation of the Curve

This part of the problem shifts gears a bit. We're now tasked with finding the equation of a curve, given some information about its derivative and a point it passes through. This is a classic differential equation problem, and we'll use integration to find the curve's equation. The problem statement wasn't fully provided, so we will use a common example and make some assumption to demostrate the steps of solving this type of questions.

Let's assume we are given that the curve passes through the point (1, 3) and its derivative is given by dy/dx = 10x√(x^2 + 8). Our goal is to find the equation y = f(x) that satisfies these conditions. This involves solving a differential equation, which means we need to find the antiderivative of the given derivative.

  1. Separate variables (if necessary): In this case, the variables are already separated, with 'dy' on the left and 'dx' on the right.

  2. Integrate both sides: We integrate both sides of the equation dy/dx = 10x√(x^2 + 8) with respect to x:

    ∫dy = ∫10x√(x^2 + 8) dx

    The left side is straightforward: ∫dy = y + C1, where C1 is a constant of integration.

    For the right side, we'll use U-substitution. Let u = x^2 + 8, then du = 2x dx, so 5du = 10x dx. The integral becomes:

    ∫10x√(x^2 + 8) dx = ∫5√u du = 5∫u^(1/2) du

  3. Solve the integral: Using the power rule for integration:

    5∫u^(1/2) du = 5 * (2/3)u^(3/2) + C2 = (10/3)u^(3/2) + C2

  4. Substitute back: Substitute u = x^2 + 8 back into the expression:

    (10/3)u^(3/2) + C2 = (10/3)(x^2 + 8)^(3/2) + C2

  5. Combine constants: We now have y + C1 = (10/3)(x^2 + 8)^(3/2) + C2. We can combine the constants C1 and C2 into a single constant C: y = (10/3)(x^2 + 8)^(3/2) + C

  6. Use the given point to find C: We know the curve passes through (1, 3), so we substitute x = 1 and y = 3 into the equation to solve for C:

    3 = (10/3)(1^2 + 8)^(3/2) + C 3 = (10/3)(9)^(3/2) + C 3 = (10/3)(27) + C 3 = 90 + C C = -87

  7. Write the final equation: Substitute C = -87 back into the equation:

    y = (10/3)(x^2 + 8)^(3/2) - 87

Therefore, the equation of the curve is y = (10/3)(x^2 + 8)^(3/2) - 87. This equation satisfies the given derivative and passes through the point (1, 3). Remember, this was based on our assumption of the derivative and the point. If the original problem provided different information, the steps would be the same, but the calculations and final equation would be different.

Conclusion

We've successfully integrated ∫x√(5x-2) dx using both U-substitution and integration by parts, and we've shown how the results, though initially different in appearance, are indeed equivalent. We also tackled a curve equation problem, reinforcing how integration helps us move from derivatives back to the original functions. Keep practicing these techniques, guys, and you'll become integration masters in no time! Remember, calculus is a journey, and every problem you solve makes you a stronger mathematician.