Introduction

Welcome to this guide on Indefinite Integral, this article will guide you through the following:

What is Indefinite integral

Recall the Fundamental Theorem of Calculus:

${\int }_a^{b}f(x)dx=F(b)-F(a)$

Where $F^{\prime }(x)=f(x)$. If we remove the integration bound of the integral, the newly defined operation is called indefinite integral: given a function $f(x)$, it returns a function which 

has a derivative of $f(x)$, we denote this operation as this:

$\int f(x)dx=F(x)$

Where $F^{\prime }(x)=f(x)$.

However this is not the complete result, recall that the derivative of a constant is $0$, which means $\frac{d}{dx}F(x)+C=f(x)$, thus for an indefinite integral, the correct answer would be

$\int f(x)dx=F(x)+C$

Where $C$ is called the Constant of Integration and $F^{\prime }(x)=f(x)$, please do not forget to add this constant in indefinite integral, CollegeBoard will take off marks if you forgot it.

Evaluating Indefinite Integral

Below is a list of basic integrals, most integrals can be reduced (through methods introduced later in the course) into these forms, it is highly recommended that you memorize all the integrals in the list.

Indefinite Integral also follows two important rules:

Example 1: Evaluate $\int (\cos x+7e^x)dx$

We can split this integral into 2 and evaluate them separately:

$\int (\cos x+7e^x)dx=\int \cos xdx+\int 7e^{x}dx$

Recall the definition of indefinite integral, if we want to find the indefinite integral of

$\cos x$, we are finding the antiderivative of $\@cos x$, which is $\sin x$, thus we have:

$\int \cos xdx=\sin x+C_1$

Same logic apply for $7e^x$, we have

$\int 7e^{x}dx=7e^x+C_2$

Sum everything up, we have

$\int (\cos x+7e^x)dx=\sin x+C_1+7e^x+C_2$

Here $C_1$ and $C_2$ are simply constant, we denote their sum as a new constant $C$, thus we have

$\int (\cos x+7e^x)dx=\sin x+7e^x+C$

Do not forget the Constant of Integration, to check if the answer is correct, take the derivative of the result and see if it matches with the integrand.

Example 2: Evaluate $\int \left(\frac{1}{\sqrt[3]{x^2}}-\frac{4}{5x}+\frac{8}{1+x^2}\right)dx$

To evaluate this integral, split it as a sum of 3 integrals, let $I=\int \left(\frac{1}{\sqrt[3]{x^2}}-\frac{4}{5x}+\frac{8}{1+x^2}\right)dx$:

$=\int \frac{1}{\sqrt[3]{x^2}}dx-\int \frac{4}{5x}dx+\int \frac{8}{1+x^2}dx$

$I=\int x^{-2/3}dx-\frac{4}{5}\int \frac{1}{x}dx+8\int \frac{1}{1+x^2}dx$

$=3x^{\frac{1}{3}}-\frac{4}{5}\ln |x|+8\arctan x+C$

Example 3: Evaluate $\int {\tan }^{2}xdx$

This integral is not in the list of integrals shown above, however it can be turned into one. Recall that ${\tan }^{2}x+1={\sec }^{2}x$, we have

$=\int ({\sec }^{2}x-1)dx$

$=\int {\sec }^{2}xdx-\int dx$

$=\tan x-x+C$

It may be tempting to assume that all functions are “integrateable” (the antiderivative can be expressed in elementary functions) after all the integrals seen. However that is not the case, not all functions have an antiderivative that can be expressed with elementary functions. The most famous example is the bell curve in statistics, $(e^{-x}{)}^2$, this function is not “integrable”. Other examples include: $\sqrt{1-x^4}$, which is known as the elliptical function (it arises from calculating the circumference of ellipse), $\sin (x^2)$, which is known as the Fresnel integral. To deal with such integrals, using a numeric method is the optimal choice.

Practice