Integrating Trigonometric Functions by Double Angle Formula Integrating Trigonometric Functions can be done by Double Angle Formula reducing the power of trigonometric functions. \( \begin{aligned} \displaystyle \cos{2A} &= 2\cos^2{A} – 1 \\ &= 1 – 2\sin^2{A} \\ &= \cos^2{A} – \sin^2{A} \\ \end{aligned} \\ \) Practice Questions Question 1 Find \( \displaystyle \int{\cos^2{x}}dx \). Question […]

# Integration

# Trigonometric Integration by Substitution

Trigonometric Integration by Substitution Trigonometric Integration by Substitution can be handled using the basic rules such as \( \displaystyle \frac{d}{dx} \sin{x} = \cos{x} \text{ and } \frac{d}{dx} \cos{x} = -\sin{x} \). Practice Questions of Trigonometric Integration by Substitution Question 1 Find \( \displaystyle \int{\sin{x}\cos^2{x}}dx \). Question 2 Find \( \displaystyle \int{\cos{x}\sin^2{x}}dx \). Question 3 Find […]

# Definite Integration by Substitution

Definite Integration by Substitution Definite Integration by Substitution requires to convert upper and lower limits of definite integration. $$ \displaystyle \int_{x=a}^{x=b}{f(x)}dx = \int_{u=c}^{u=d}{f(u)}du $$ Practice Questions of Definite Integration by Substitution Question 1 Find \( \displaystyle \int_{-1}^{0}{x(1+x)^{10}}dx \). Question 2 Find \( \displaystyle \int_{0}^{1}{x \sqrt{1-x^2}}dx \). Question 3 Find \( \displaystyle \int_{3}^{18}{\frac{x}{\sqrt{x-2}}}dx \).

# Integration by Substitution

Integration by Substitution Integration by Substitution is performed by replacing the pronumeral (variable) to other pronumeral to simplify the expression for integrating easier. Students often made mistakes by forgetting to replace back to the original pronumeral. Practice Questions of Integration by Substitution Question 1 Find \( \displaystyle \int{\frac{2x}{\sqrt{x^2-4}}}dx \). Question 2 Find \( \displaystyle \int{\frac{x^2}{\sqrt{1-x^3}}}dx […]

# Definite Integrals

Background of Definite Integrals Definite Integrals are calculated by subtracting the function values of upper and lower limits. $$\displaystyle \int_{a}^{b}{f(x)}dx = F(a) – F(b), \text{where } \frac{d}{dx}F(x) = f(x)$$ Practice Questions Question 1 Evaluate \( \displaystyle \int_{0}^{2}{6x}dx \). Question 2 Evaluate \( \displaystyle \int_{2}^{4}5dx\). Question 3 Evaluate \( \displaystyle \int_{1}^{3}{(2x+4)}dx\). Question 4 Evaluate \( \displaystyle […]

# Indefinite Integral of Rational Functions

Understanding Indefinite Integral of Rational Functions Using Indefinite Integral of Rational Functions requires that the format of the expression must be power of linear expressions, such as \( (3x-1)^3, (2x+3)^3, \sqrt{4x-1} \), etc. \( (3x^2-1)^3, (2\sqrt{x}+3)^3, \sqrt{4x^3-1} \) are not applicable for this formula. $$\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + C \ (n \ne -1)$$ Practice […]

# Radical Indefinite Integrals

Radical Indefinite Integrals should be performed after converting its radical or surd notations into index form. $$\displaystyle \sqrt[n]{x^m} = x^{\frac{m}{n}}$$ Practice Questions Question 1 Find \( \displaystyle \int{\sqrt{x}}dx \). Question 2 Find \( \displaystyle \int{\sqrt[3]{x^5}}dx \). Question 3 Find \( \displaystyle \int{\sqrt{5x^3}}dx \). Question 4 Find \( \displaystyle \int{\frac{3x-1}{\sqrt{x}}}dx \).

# Indefinite Integral Formula

Basics of Indefinite Integral Formula Indefinite Integral Formula ahs been made from the reverse operations of differentiation or anti-differentiation. \( \begin{aligned} \displaystyle \frac{d}{dx}x^3 &= 3x^2 &\Rightarrow \int{3x^2}dx &= x^3 \\ \frac{d}{dx}(x^3 + 4) &= 3x^2 &\Rightarrow \int{3x^2}dx &= x^3 + 4 \\ \frac{d}{dx}(x^3 -2) &= 3x^2 &\Rightarrow \int{3x^2}dx &= x^3 – 2 \\ \end{aligned} \\ […]

# Integration Reverse Chain Rule

By recalling the chain rule, Integration Reverse Chain Rule comes from the usual chain rule of differentiation. This skill is to be used to integrate composite functions such as \( e^{x^2+5x}, \cos{(x^3+x)}, \log_{e}{(4x^2+2x)} \). Let’s take a close look at the following example of applying the chain rule to differentiate, then reverse its order to […]

# Integration by Parts

Integration by Parts is made of product rule of differentiation. The derivative of \(uv\) is \(u’v + uv’\) and integrate both sides. \( \begin{aligned} \require{color} (uv)’ &= u’v + uv’ \\ uv &= \int u’v + \int uv’ \\ \int u’v &= uv – \int uv’ \text{ or } \int uv’ = uv – \int […]

# Integration Recurrence Formula

For Integration Recurrence Formula or reduction formula, it is important set a relationship between two consecutive terms by using mostly the integration by parts. Worked Example of Integration Recurrence Formula Question 1 For any integer \(m \ge 0\) let \(\displaystyle I_m = \int_{0}^{1} x^m (x^2 – 1)^5 dx \). Prove that for \( \displaystyle m\ge […]

# Integration using Trigonometric Properties

Trigonometric properties such as the sum of squares of sine and cosine with the same angle is one, $$ \displaystyle \sin^2{\theta} + \cos^2{\theta} = 1 \\ \cos\Big(\frac{\pi}{2} – \theta \Big) = \sin{\theta} $$ can simplify harder integration. Worked Example of Integration using Trigonometric Properties (a) Find \(a\) and \(b\) for \(\displaystyle \frac{1}{x(4-x)} = \frac{a}{x} + […]