Problem

Integrate Using u-Substitution integral of x/( square root of 1-x^2) with respect to x

The problem is asking for the evaluation of an integral using the method of u-substitution. The integral in question is the integral of the function x/(√(1-x^2)) with respect to x. To approach this integral with u-substitution, you typically look for a function inside the integral that, when differentiated, appears elsewhere in the integral. The integral suggests a potential substitution with u being related to the expression (1-x^2) because its derivative, -2x, is related to the x in the numerator. The goal is to rewrite the integral in terms of u to simplify the integration process. The u-substitution method is a common technique used to evaluate more complex integrals by changing variables to simplify the integral before performing the integration.

x1x2dx

Answer

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Solution:

Step:1

Choose u=1x2. Consequently, we have du=2xdx, which implies 12du=xdx. Now express the integral in terms of u and du.

Step:1.1

Set u=1x2 and compute dudx.

Step:1.1.1

Take the derivative of 1x2: ddx[1x2].

Step:1.1.2

Proceed with differentiation.

Step:1.1.2.1

Using the Sum Rule, the derivative of 1x2 is the sum of the derivatives of each term: ddx[1]+ddx[x2].

Step:1.1.2.2

The derivative of a constant, 1, is 0: 0+ddx[x2].

Step:1.1.3

Evaluate the derivative of x2: ddx[x2].

Step:1.1.3.1

Applying the constant multiple rule, the derivative of x2 is 1 times the derivative of x2: 0ddx[x2].

Step:1.1.3.2

Apply the Power Rule, which states that the derivative of xn is nxn1, where n=2: 0(2x).

Step:1.1.3.3

Combine the constant 1 with 2: 02x.

Step:1.1.4

Subtract 2x from 0: 2x.

Step:1.2

Express the integral in terms of u and du: 1u12du.

Step:2

Begin simplification.

Step:2.1

Extract the negative sign from the fraction: 1u(12)du.

Step:2.2

Combine 1u with 12: 1u2du.

Step:2.3

Rearrange the fraction: 12udu.

Step:3

Extract the constant 1 from the integral: 12udu.

Step:4

Extract the constant 12 from the integral: (121udu).

Step:5

Apply exponent rules.

Step:5.1

Express u as u12: 121u12du.

Step:5.2

Rewrite u12 in the denominator as u12: 12u12du.

Step:5.3

Simplify the exponent.

Step:5.3.1

Apply the power of a power rule: 12u121du.

Step:5.3.2

Combine the exponents: 12u12du.

Step:5.3.3

Keep the negative sign outside the fraction: 12u12du.

Step:6

Integrate using the Power Rule: 12(2u12+C).

Step:7

Final simplification.

Step:7.1

Multiply 12 by 2u12: u12+C.

Step:7.2

Combine terms: u12+C.

Step:8

Substitute back u with 1x2: ((1x2)12)+C.

Knowledge Notes:

The u-substitution method is a technique used in calculus to simplify the process of integration by substituting a part of the integrand with a new variable, typically denoted as u. This method often simplifies the integral into a form that is easier to integrate.

The steps involved in u-substitution generally include:

  1. Identifying a part of the integrand to substitute with u.

  2. Differentiating u with respect to x to find du/dx and solving for dx.

  3. Substituting u and dx into the integral.

  4. Simplifying the integral if possible.

  5. Integrating with respect to u.

  6. Substituting back the original variable x using the initial substitution.

The Power Rule for integration states that the integral of xn with respect to x is xn+1n+1+C, provided n1. When n=1, the integral is ln|x|+C.

The Sum Rule for differentiation states that the derivative of a sum of functions is the sum of their derivatives.

The Constant Multiple Rule allows us to take constants out of the derivative or integral.

The Power Rule for differentiation states that the derivative of xn with respect to x is nxn1.

These rules are fundamental to calculus and are used extensively in solving integration and differentiation problems.

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