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Calculate the following indefinite integrals by using linearity, the power rule, and the tables of antiderivatives. (12) ∫4x+x2dx ∫x23​+x25​dx ∫1+x21​+x2dx

Answer :

Final answer:

The indefinite integrals are:

  1. ∫4x+x^2dx = 2x^2 + (1/3)x^3 + C
  2. ∫x^2/3 + x^2/5dx = x^(5/3) + x^(7/5) + C
  3. ∫1+x^2/1+x^2dx = x + arctan(x) + C

Explanation:

To calculate the indefinite integrals, we will use linearity, the power rule, and the tables of antiderivatives.

For the first integral, ∫4x+x^2dx:

  1. Using linearity, we can split the integral into two separate integrals: ∫4xdx + ∫x^2dx.
  2. Applying the power rule, the integral of 4x with respect to x is (4/2)x^2 = 2x^2. The integral of x^2 with respect to x is (1/3)x^3.
  3. Combining the results, the indefinite integral of 4x+x^2dx is 2x^2 + (1/3)x^3 + C, where C is the constant of integration.

For the second integral, ∫x^2/3 + x^2/5dx:

  1. Using linearity, we can split the integral into two separate integrals: ∫x^2/3dx + ∫x^2/5dx.
  2. Applying the power rule, the integral of x^2/3 with respect to x is (3/3)x^(2/3+1) = (3/3)x^(5/3) = x^(5/3). The integral of x^2/5 with respect to x is (5/5)x^(2/5+1) = (5/5)x^(7/5) = x^(7/5).
  3. Combining the results, the indefinite integral of x^2/3 + x^2/5dx is x^(5/3) + x^(7/5) + C, where C is the constant of integration.

For the third integral, ∫1+x^2/1+x^2dx:

  1. Using linearity, we can split the integral into two separate integrals: ∫1dx + ∫x^2/1+x^2dx.
  2. The integral of 1 with respect to x is x.
  3. For the second integral, we can simplify the integrand by factoring out x^2: x^2/1+x^2 = x^2(1/(1+x^2)).
  4. Using a table of antiderivatives, we find that the integral of 1/(1+x^2) with respect to x is arctan(x).
  5. Combining the results, the indefinite integral of 1+x^2/1+x^2dx is x + arctan(x) + C, where C is the constant of integration.

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