Indefinite integrals

In a nutshell

Indefinite integrals are integrals that have a constant of integration. Indefinite integrals can be represented with an integration symbol.

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The integration symbol

The integration symbol is $$\int$$​. This is used to integrate functions as follows:

Note: $$dx$$ can be read as "with respect to $$x$$".

Examples

Evaluate the following integrals:

i) $$\int x^3\,dx$$​

ii) $$\int (3\sqrt{x}+\dfrac{1}{x^2})\,dx$$​

iii) $$\int (6y^2 +2ky+5)\,dy$$

Part i):

Integrate $$x^3$$ using the rule:

$$\underline{\int x^3 \, dx = \dfrac{x^4}{4}+C}$$​​

Part ii):

Write each term in the form $$ax^n$$ and integrate them separately:

$$\begin{aligned}\int (3\sqrt{x}+\dfrac{1}{x^2}) \, dx &= \int (3x^{\frac{1}{2}}+x^{-2})\,dx \\ &= \int 3x^{\frac12} \, dx + \int x^{-2} \, dx \\ &=\dfrac{3}{(\frac{3}{2})}x^{\frac{3}{2}}+\dfrac{1}{-1}x^{-1}+C\\ &= 2x^{\frac{3}{2}}-x^{-1}+C \end{aligned}$$​​

​$$\underline{\int (3\sqrt{x}+\dfrac{1}{x^2}) \, dx=2x^{\frac{3}{2}}-x^{-1}+C}$$​​

Part iii):

The $$dy$$ means to integrate with respect to $$y$$.

$$\begin{aligned}\int (6y^2 +2ky+5)\,dy &= \int 6y^2 \, dy + \int 2ky^1 \, dy + \int 5y^0\, dy \\&=\dfrac{6}{3}y^3+\dfrac{2k}{2}y^2+\dfrac{5}{1}y^1+C\\&=2y^3+ky^2+5y+C\end{aligned}$$​​

​$$\underline{\int (6y^2 +2ky+5)\,dy=2y^3+ky^2+5y+C}$$​​