Powers and linearity

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Split sums and move constant factors outside. For each power, add one to the exponent and divide by the new exponent. The exception is x⁻¹: its primitive is ln|x|. Always check by differentiating and add a single constant C to the final result.

Worked example

1. Identify the integrand, variables and order. For multiple integrals, start with the innermost integral.

   $$\int \left(3 x^{2} - 4 x + 2\right)\, dx$$

2. Integrate with respect to x. Treat the other variables as constants at this stage.

   $$\int \left(3 x^{2} - 4 x + 2\right)\, dx$$

3. Split the sum. The integral of a sum is the sum of its integrals.

   $$\int \left(3 x^{2} - 4 x + 2\right)\, dx = \int 2\, dx + \int \left(- 4 x\right)\, dx + \int 3 x^{2}\, dx$$

4. Move the constant factor outside: it does not depend on the integration variable.

   $$\int 3 x^{2}\, dx = x^{3}$$

5. Apply the power rule: add 1 to the exponent and divide by the new exponent. This requires n ≠ −1.

   $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

6. Combine the results of the substeps and simplify.

   $$\int 3 x^{2}\, dx = x^{3}$$

7. Move the constant factor outside: it does not depend on the integration variable.

   $$\int \left(- 4 x\right)\, dx = - 2 x^{2}$$

8. Apply the power rule: add 1 to the exponent and divide by the new exponent. This requires n ≠ −1.

   $$\int x\, dx = \frac{x^{2}}{2}$$

9. Combine the results of the substeps and simplify.

   $$\int \left(- 4 x\right)\, dx = - 2 x^{2}$$

10. Integrate a constant by multiplying it by the variable.

    $$\int 2\, dx = 2 x$$

11. Combine the results of the substeps and simplify.

    $$\int \left(3 x^{2} - 4 x + 2\right)\, dx = x^{3} - 2 x^{2} + 2 x$$

12. Check this primitive by differentiating it: recover exactly the integrand of this stage.

    $$3 x^{2} - 4 x + 2 = 3 x^{2} - 4 x + 2$$

13. Add the constant C: all primitives on an interval differ by a constant.

    $$x \left(x^{2} - 2 x + 2\right) + C$$