Problem

Integrate Using u-Substitution integral of 28(7x-2)^3 with respect to x

The problem is asking to perform an integration using the u-substitution method. This involves identifying a part of the integrand that can be substituted with a new variable, typically denoted as 'u', which simplifies the integration process. In the expression provided, the goal is to integrate a function of the form 28(7x-2)^3 with respect to x by substituting the inner function 7x-2 as 'u' to make the integral easier to evaluate.

28((7x2))3dx

Answer

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Solution:

Step:1

Assign u=7x2. Consequently, du=7dx, leading to the expression dx=17du. Proceed by substituting u and du into the integral.

Step:1.1

Set u=7x2 and compute dudx.

Step:1.1.1

Take the derivative of 7x2 with respect to x: ddx(7x2).

Step:1.1.2

Apply the Sum Rule in differentiation to find the derivative of 7x2 as ddx(7x)+ddx(2).

Step:1.1.3

Determine ddx(7x).

Step:1.1.3.1

Given that 7 is a constant, the derivative of 7x is 7ddx(x).

Step:1.1.3.2

Utilize the Power Rule, which states that ddx(xn)=nxn1 for n=1, to differentiate x.

Step:1.1.3.3

Multiply 7 by 1 to get 7.

Step:1.1.4

Employ the Constant Rule for differentiation.

Step:1.1.4.1

Since 2 is a constant, its derivative with respect to x is 0.

Step:1.1.4.2

Combine 7 and 0 to obtain 7.

Step:1.2

Transform the integral into terms of u and du: 4u3du.

Step:2

Extract the constant 4 from the integral: 4u3du.

Step:3

Apply the Power Rule for integration to find the antiderivative of u3 as 14u4: 4(14u4+C).

Step:4

Simplify the expression.

Step:4.1

Express 4(14u4+C) as 4(14)u4+C.

Step:4.2

Rewrite 4(14)u4+C as u4+C.

Step:4.3

Multiply u4 by 1 to maintain the expression as u4+C.

Step:5

Substitute back the original variable: (7x2)4+C.

Knowledge Notes:

The u-substitution method is a technique used in calculus to simplify the process of integration, particularly when dealing with composite functions. It involves the following steps:

  1. Choosing u: Identify a part of the integrand that can be substituted with a new variable u, which simplifies the integral.

  2. Differentiating u: Compute dudx to find the relationship between dx and du.

  3. Substituting: Replace all instances of the chosen part of the integrand with u and dx with the corresponding du expression.

  4. Integrating: Perform the integration with respect to u.

  5. Back-substitution: Replace u with the original expression to return to the variable x.

  6. Simplification: Simplify the resulting expression if possible.

The Power Rule for integration states that xndx=1n+1xn+1+C for any real number n1, where C is the constant of integration.

The Sum Rule for differentiation states that the derivative of a sum of two functions is the sum of their derivatives: ddx(f(x)+g(x))=dfdx+dgdx.

The Constant Rule for differentiation states that the derivative of a constant is zero: ddx(c)=0 for any constant c.

By applying these rules and the process of u-substitution, complex integrals can be evaluated more easily.

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