Integration By Parts - Rules, Formula and Example

Integration by Parts Formula

The formula for integrating by parts is:

\( \int u \space dv = uv – \int v \space du \)

Where,
u = function of u(x)
dv = variable dv
v = function of v(x)
du = variable du

Definite Integral

A Definite Integral has start and end values, forming an interval [a, b].

a and b (also known as limits) are placed at the bottom and top of the “S” as follows:

Integration Rules of Basic Functions

1- Integration of Constant

The constant of integration is the constant ‘C’ multiplied by the integration result. The constant of integration represents the term of the original expression that cannot be obtained from the function’s antiderivative.

\( \frac{d}{dx} f(x) = f'(x) \)

\( \int f′(x) \space dx = g(x) + C \)

And \( g(x) + C = f(x)\).

2- Integration of Variable

If x is any variable, then integration of the variable is;

\( \int x \space dx = \dfrac{x^2}{2} + C \)

3- Integration of Square

If the given function is a square term, then integration can find out using this formula:

\( \int x^2 dx = \dfrac{x^3}{3} \)

4- Integration of Reciprocal

The reciprocal rule of integration states that the integral of a variable’s multiplicative inverse equals the sum of the variable’s natural logarithm and the constant of integration.

\( \int (\dfrac{1}{x}) dx = ln|x| + C \)

5- Integration of Exponential Function

Below are the different rules for the integration of exponential functions are:

\( \int e^x dx = e^x + C \)

\( \int a^x dx = \dfrac{a^x}{ln \space a} + C \)

6- Integration of Trigonometric Function

We can find the integration of trigonometric functions using these formulas:

• \(\int\)sin x dx = -cos x + C
• \(\int\)cos x dx = sin x + C
• \(\int\)tan x dx = ln|sec x| + C

Rules Of Integration

The crucial rules for integration are:

1- Power Rule

The power rule for integration gives us a formula for integrating any function written as a power of x. 

\( \int x^n \ dx = \dfrac{x^{n+1}}{n+1} + C \)

Example: Integrate \( \int x^4 dx \)

\( \int x^4 dx =  \dfrac{x^{4+1}}{4+1} = \dfrac{x^5}{5} \)

2- Multiplication By Constant

If a function is multiplied by a constant, then its integration is given by:

\( \int cf(x) \space dx = c \int f(x) \space dx \)

Example: Integrate \( \int -5 x^6 dx \)

\( = – 5((\dfrac{1}{6+1}) \space x^{6+1} + C) \)

\( = – 5 (\frac{1}{7} \space x^7 + C) \)

\( = \frac{-5}{7} \space x^7 + C \)

3- Sum Rule

The sum rule of integration states that the integral of the sum of two functions equals the sum of individual function integration.

\( \int(f + g) \space dx = \int f \space dx + \int g \space dx \)

4- Difference Rule

The difference between any two functions’ integrals is equal to the difference between their integrals. The difference rule of integration is a mathematical property expressed as an equation.

\( \int(f – g) \space dx = \int f \space dx – \int g \space dx \)

5- Substitution Rule

Any given integral is transformed into a simple form of integral using this method of integration by substitution by substituting other variables for the independent variable. For example, the following is the General Form of integration by substitution: 

\( \int f(g(x)).g'(x).dx = f(t).dt\), where \(t = g(x) \)

6- Chain Rule Integration

The chain rule is a technique for calculating the derivative of a composite function. Integration is not possible with the chain rule, but reversing the chain rule results in integration by substitution.

\( \int f(g(x))g'(x)dx = f(g(x)) + C \)

7- Quotient Rule Integration

The integral quotient rule is a method for integrating two functions with the numerator and denominator values. 

The Integral Division rule formula is derived from the Integration by Parts \(\frac{u}{v}\) formula.

\( \int udv = u.v = \int v.du \)

How to do Integration by parts

Example 1 : Integrate \( \int \dfrac{1}{x^7} dx \)

\( \int (\dfrac{1}{x^7}) dx = \int x^{-7} dx  \)

\( = \dfrac{x^{-7 + 1}}{-7 + 1} + C \)

\( = \dfrac{x^{-6}}{-6} + C \)

\( = \dfrac{-1}{6x^6} + C \)

Example 2: Integrate x sin2x 

We use the integration by parts formula:

\( \int uv.dx = u\int v.dx – \int( u’ \int v.dx).dx \)

Here u = x, 

v = Sin2x

\(\int\) x sin2x. dx

\( = x \int sin2x \space dx – \frac{d}{dx}. x. \int sin2x \space dx. dx \)

\( = x. -cos(\frac{2x}{2}) – \int (1.-cos(\frac{2x}{2})). dx \)

\( = -cos(\frac{2x}{2}) . dx + \frac{1}{2} cos2x \space dx \)

\( = -x cos(\frac{2x}{2}) + sin(\frac{2x}{4}) + C \)

Thu \( \int x sin2x \space dx = -x cos(\frac{2x}{2}) +sin(\frac{2x}{4}) + C \)

Example 3: Integrate \( \int (1 – x^2) – \frac{1}{2} dx \)

\( \int (1 – x^2) – \frac{1}{2} dx = \int \frac{1}{1 – x^2} \frac{1}{2} dx \)

\( = \int \dfrac{1}{\sqrt{1 – x^2}} dx \)

\( = sin^{-1} x + c\)

Where did “Integration by parts” come from?

Integration by parts can be considered an integral version of the product rule of differentiation.

(uv)’ = uv’ + u’v

Integrate both sides and rearrange:

\(\int\)(uv)’ dx = \(\int\)uv’ dx + \(\int\)u’v dx

uv = \(\int\)uv’ dx + \(\int\)u’v dx

\(\int\)uv’ dx = uv − \(\int\)u’v dx

Key Takeaways

• Integration by parts does not apply to all functions. It does not, for example, work for \(\int \sqrt{x}\) sin x dx because there is no function whose derivative is \(\sqrt{x}\) sin x.
• When calculating the integral of the second function, do not include the integration constant. This can be done only once, at the end of the integration process. Even if you add it, it will not change the answer but complicate the calculation.
• Typically, we take the first function if it is a power of x or a polynomial in x. When another function is an inverse trigonometric or logarithmic function, we use it as the first function.