Integration Formulas Guide: Basic to Intermediate with Examples

Complete Integration Formulas: Basic to Intermediate

This guide covers every essential integration formula from fundamentals to intermediate techniques, with worked examples and practice problems.

1. Basic Integration Rules

1.1 Fundamental Rules

Rule	Formula
Constant Rule	$\displaystyle \int k \, dx = kx + C$
Power Rule	$\displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where
Constant Multiple	$\displaystyle \int k f(x) \, dx = k \int f(x) \, dx$
Sum/Difference	$\displaystyle \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$

1.2 Example: Power Rule

Problem: Evaluate $\displaystyle \int 3x^4 \, dx$

Solution:

\int 3x^4 \, dx = 3 \cdot \frac{x^{4+1}}{4+1} + C = 3 \cdot \frac{x^5}{5} + C = \frac{3x^5}{5} + C

2. Standard Algebraic Integrals

\int dx = x + C

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

\int \frac{1}{x} \, dx = \ln|x| + C

\int e^x \, dx = e^x + C

\int a^x \, dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)

2.1 Example: Exponential and Logarithmic

Problem: Evaluate $\displaystyle \int \frac{5}{x} \, dx + \int 2e^x \, dx$

Solution:

\int \frac{5}{x} \, dx + \int 2e^x \, dx = 5\ln|x| + 2e^x + C

3. Trigonometric Integrals

3.1 Basic Trigonometric Formulas

\int \sin x \, dx = -\cos x + C

\int \cos x \, dx = \sin x + C

\int \tan x \, dx = -\ln|\cos x| + C = \ln|\sec x| + C

\int \cot x \, dx = \ln|\sin x| + C

\int \sec x \, dx = \ln|\sec x + \tan x| + C

\int \csc x \, dx = \ln|\csc x - \cot x| + C = -\ln|\csc x + \cot x| + C

3.2 Square Terms

\int \sec^2 x \, dx = \tan x + C

\int \csc^2 x \, dx = -\cot x + C

\int \sec x \tan x \, dx = \sec x + C

\int \csc x \cot x \, dx = -\csc x + C

3.3 Example: Trigonometric Integration

Problem: Evaluate $\displaystyle \int (2\sin x + 3\cos x) \, dx$

Solution:

\int (2\sin x + 3\cos x) \, dx = -2\cos x + 3\sin x + C

4. Inverse Trigonometric Integrals

\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1} x + C

\int \frac{-1}{\sqrt{1-x^2}} \, dx = \cos^{-1} x + C

\int \frac{1}{1+x^2} \, dx = \tan^{-1} x + C

\int \frac{-1}{1+x^2} \, dx = \cot^{-1} x + C

\int \frac{1}{x\sqrt{x^2-1}} \, dx = \sec^{-1} x + C

\int \frac{-1}{x\sqrt{x^2-1}} \, dx = \csc^{-1} x + C

4.1 Generalized Forms

\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C

\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C

\int \frac{1}{x\sqrt{x^2-a^2}} \, dx = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C

4.2 Example: Inverse Trigonometric

Problem: Evaluate $\displaystyle \int \frac{1}{9+x^2} \, dx$

Solution: Here $a^2 = 9$ , so $a = 3$ :

\int \frac{1}{9+x^2} \, dx = \frac{1}{3}\tan^{-1}\left(\frac{x}{3}\right) + C

5. Hyperbolic Integrals

\int \sinh x \, dx = \cosh x + C

\int \cosh x \, dx = \sinh x + C

\int \tanh x \, dx = \ln(\cosh x) + C

\int \text{sech}^2 x \, dx = \tanh x + C

\int \text{csch}^2 x \, dx = -\coth x + C

6. Integration by Substitution

6.1 Method

If an integral has the form $\displaystyle \int f(g(x)) \cdot g'(x) \, dx$ , substitute $u = g(x)$ , so .

6.2 Common Substitutions

Integral Form	Substitution
$\int f(ax+b) \, dx$	$u = ax+b$

6.3 Example 1: Linear Substitution

Problem: Evaluate $\displaystyle \int (3x+2)^5 \, dx$

Solution: Let $u = 3x+2$ , then $du = 3 \, dx$ , so $dx = \frac{du}{3}$ .

\int (3x+2)^5 \, dx = \int u^5 \cdot \frac{du}{3} = \frac{1}{3} \cdot \frac{u^6}{6} + C = \frac{(3x+2)^6}{18} + C

6.4 Example 2: Logarithmic Type

Problem: Evaluate $\displaystyle \int \frac{2x}{x^2+1} \, dx$

Solution: Let $u = x^2+1$ , then $du = 2x \, dx$ .

\int \frac{2x}{x^2+1} \, dx = \int \frac{du}{u} = \ln|u| + C = \ln(x^2+1) + C

6.5 Example 3: Trigonometric Substitution

Problem: Evaluate $\displaystyle \int \sin^3 x \cos x \, dx$

Solution: Let $u = \sin x$ , then $du = \cos x \, dx$ .

\int \sin^3 x \cos x \, dx = \int u^3 \, du = \frac{u^4}{4} + C = \frac{\sin^4 x}{4} + C

7. Integration by Parts

7.1 Formula

\int u \, dv = uv - \int v \, du

Or equivalently:

\int f(x) g'(x) \, dx = f(x)g(x) - \int f'(x)g(x) \, dx

7.2 LIATE Rule (Priority for choosing $u$ )

Choose $u$ in this order of priority:

L — Logarithmic functions ( $\ln x$ )
I — Inverse trigonometric ( $\sin^{-1} x$ , $\tan^{-1} x$ )

7.3 Example 1: Algebraic × Exponential

Problem: Evaluate $\displaystyle \int x e^x \, dx$

Solution: Let $u = x$ (algebraic, priority over exponential) and $dv = e^x \, dx$ .

Then $du = dx$ and $v = e^x$ .

\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C = e^x(x-1) + C

7.4 Example 2: Logarithmic

Problem: Evaluate $\displaystyle \int \ln x \, dx$

Solution: Let $u = \ln x$ and $dv = dx$ .

Then $du = \frac{1}{x} \, dx$ and $v = x$ .

\int \ln x \, dx = x\ln x - \int x \cdot \frac{1}{x} \, dx = x\ln x - \int dx = x\ln x - x + C

7.5 Example 3: Repeated Integration by Parts

Problem: Evaluate $\displaystyle \int x^2 \sin x \, dx$

Solution: First application: $u = x^2$ , $dv = \sin x \, dx$

$du = 2x \, dx$ , $v = -\cos x$

\int x^2 \sin x \, dx = -x^2\cos x + \int 2x\cos x \, dx

Second application for $\int 2x\cos x \, dx$ :

$u = 2x$ , $dv = \cos x \, dx$
$du = 2 \, dx$ ,

\int 2x\cos x \, dx = 2x\sin x - \int 2\sin x \, dx = 2x\sin x + 2\cos x

Final answer:

\int x^2 \sin x \, dx = -x^2\cos x + 2x\sin x + 2\cos x + C

8. Integration of Rational Functions (Partial Fractions)

8.1 Types of Partial Fractions

Form of Factor in Denominator	Form of Partial Fraction
Non-repeated linear: $(ax+b)$	$\displaystyle \frac{A}{ax+b}$

8.2 Example 1: Distinct Linear Factors

Problem: Evaluate $\displaystyle \int \frac{3x+5}{(x-1)(x+2)} \, dx$

Solution:

\frac{3x+5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}

Multiply by $(x-1)(x+2)$ :

3x+5 = A(x+2) + B(x-1)

Set $x = 1$ : $8 = 3A \Rightarrow A = \frac{8}{3}$

\int \frac{3x+5}{(x-1)(x+2)} \, dx = \frac{8}{3}\ln|x-1| + \frac{1}{3}\ln|x+2| + C

8.3 Example 2: Repeated Linear Factor

Problem: Evaluate $\displaystyle \int \frac{x+1}{(x-1)^2} \, dx$

Solution:

\frac{x+1}{(x-1)^2} = \frac{A}{x-1} + \frac{B}{(x-1)^2}

Multiply by $(x-1)^2$ :

x+1 = A(x-1) + B

Set $x = 1$ : $2 = B$
Compare $x$ coefficients: $1 = A$ , so $A = 1$

\int \frac{x+1}{(x-1)^2} \, dx = \int \frac{1}{x-1} \, dx + \int \frac{2}{(x-1)^2} \, dx = \ln|x-1| - \frac{2}{x-1} + C

9. Trigonometric Integrals (Advanced)

9.1 Powers of Sine and Cosine

Case 1: $\int \sin^m x \cos^n x \, dx$ where $m$ is odd.

Save one $\sin x$ , convert rest to $\cos x$ using $\sin^2 x = 1-\cos^2 x$ .

Case 2: $n$ is odd.

Save one $\cos x$ , convert rest to $\sin x$ using $\cos^2 x = 1-\sin^2 x$ .

Case 3: Both $m$ and $n$ are even.

Use half-angle formulas:

\sin^2 x = \frac{1-\cos 2x}{2}, \quad \cos^2 x = \frac{1+\cos 2x}{2}

9.2 Example: Even Powers

Problem: Evaluate $\displaystyle \int \sin^2 x \, dx$

Solution:

\int \sin^2 x \, dx = \int \frac{1-\cos 2x}{2} \, dx = \frac{1}{2}\left(x - \frac{\sin 2x}{2}\right) + C = \frac{x}{2} - \frac{\sin 2x}{4} + C

9.3 Products to Sums

\int \sin(mx)\cos(nx) \, dx, \quad \int \sin(mx)\sin(nx) \, dx, \quad \int \cos(mx)\cos(nx) \, dx

Use identities:

\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]

\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]

\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]

10. Trigonometric Substitution

10.1 Standard Substitutions

Expression	Substitution	Identity Used
$\sqrt{a^2-x^2}$

10.2 Example: $\sqrt{a^2-x^2}$ Form

Problem: Evaluate $\displaystyle \int \frac{1}{\sqrt{4-x^2}} \, dx$

Solution: Let $x = 2\sin\theta$ , then $dx = 2\cos\theta \, d\theta$ .

\sqrt{4-x^2} = \sqrt{4-4\sin^2\theta} = 2\cos\theta

\int \frac{1}{\sqrt{4-x^2}} \, dx = \int \frac{2\cos\theta}{2\cos\theta} \, d\theta = \int d\theta = \theta + C = \sin^{-1}\left(\frac{x}{2}\right) + C

11. Special Algebraic Integrals

11.1 Forms with Square Roots

\int \frac{1}{\sqrt{x^2+a^2}} \, dx = \ln\left|x + \sqrt{x^2+a^2}\right| + C = \sinh^{-1}\left(\frac{x}{a}\right) + C

\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \ln\left|x + \sqrt{x^2-a^2}\right| + C = \cosh^{-1}\left(\frac{x}{a}\right) + C

\int \sqrt{x^2+a^2} \, dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln\left|x + \sqrt{x^2+a^2}\right| + C

\int \sqrt{x^2-a^2} \, dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln\left|x + \sqrt{x^2-a^2}\right| + C

\int \sqrt{a^2-x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C

11.2 Example: Special Form

Problem: Evaluate $\displaystyle \int \sqrt{x^2+9} \, dx$

Solution: Using the formula with $a = 3$ :

\int \sqrt{x^2+9} \, dx = \frac{x}{2}\sqrt{x^2+9} + \frac{9}{2}\ln\left|x + \sqrt{x^2+9}\right| + C

12. Definite Integrals

12.1 Fundamental Theorem of Calculus

If $F'(x) = f(x)$ , then:

\int_a^b f(x) \, dx = F(b) - F(a) = \left[F(x)\right]_a^b

12.2 Properties

\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx

\int_a^a f(x) \, dx = 0

\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx

\int_a^b kf(x) \, dx = k\int_a^b f(x) \, dx

12.3 Even and Odd Functions

If $f$ is even ( $f(-x) = f(x)$ ):

\int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx

If $f$ is odd ( $f(-x) = -f(x)$ ):

\int_{-a}^a f(x) \, dx = 0

12.4 Example: Definite Integral

Problem: Evaluate $\displaystyle \int_0^{\frac{\pi}{2}} \sin x \, dx$

Solution:

\int_0^{\frac{\pi}{2}} \sin x \, dx = \left[-\cos x\right]_0^{\frac{\pi}{2}} = -\cos\left(\frac{\pi}{2}\right) - (-\cos 0) = -0 + 1 = 1

13. Reduction Formulas

13.1 Wallis' Integrals

For $I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx = \int_0^{\frac{\pi}{2}} \cos^n x \, dx$ :

I_n = \frac{n-1}{n} I_{n-2}

With base cases:

$I_0 = \frac{\pi}{2}$
$I_1 = 1$

Final Formulas:

If $n$ is even:

\int_0^{\frac{\pi}{2}} \sin^n x \, dx = \frac{(n-1)!!}{n!!} \cdot \frac{\pi}{2}

If $n$ is odd:

\int_0^{\frac{\pi}{2}} \sin^n x \, dx = \frac{(n-1)!!}{n!!}

13.2 Reduction Formula for $\tan^n x$

I_n = \int \tan^n x \, dx = \frac{\tan^{n-1} x}{n-1} - I_{n-2}

13.3 Reduction Formula for $\sec^n x$

I_n = \int \sec^n x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} I_{n-2}

14. More Standard Integrals (Quick Reference)

14.1 Tangent Half-Angle (Weierstrass Substitution)

If $t = \tan\frac{x}{2}$ , then:

\sin x = \frac{2t}{1+t^2}, \quad \cos x = \frac{1-t^2}{1+t^2}, \quad dx = \frac{2 \, dt}{1+t^2}

14.2 Special Rational Forms

\int \frac{1}{x^2-a^2} \, dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C

\int \frac{1}{a^2-x^2} \, dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C

\int \frac{1}{x^2+a^2} \, dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C

\int \frac{x}{x^2+a^2} \, dx = \frac{1}{2}\ln(x^2+a^2) + C

\int \frac{x}{x^2-a^2} \, dx = \frac{1}{2}\ln|x^2-a^2| + C

15. Solved Examples (Mixed Practice)

Example 1: Substitution with Root

Problem: $\displaystyle \int \frac{x}{\sqrt{x+1}} \, dx$

Solution: Let $u = x+1$ , so $x = u-1$ and $dx = du$ .

\int \frac{u-1}{\sqrt{u}} \, du = \int (u^{\frac{1}{2}} - u^{-\frac{1}{2}}) \, du = \frac{2}{3}u^{\frac{3}{2}} - 2u^{\frac{1}{2}} + C

Substitute back:

= \frac{2}{3}(x+1)^{\frac{3}{2}} - 2(x+1)^{\frac{1}{2}} + C = \frac{2}{3}\sqrt{x+1}(x+1-3) + C = \frac{2}{3}\sqrt{x+1}(x-2) + C

Example 2: Integration by Parts — Cyclic

Problem: $\displaystyle \int e^x \sin x \, dx$

Solution: Let $I = \int e^x \sin x \, dx$ .

First parts: $u = \sin x$ , $dv = e^x dx$

$du = \cos x \, dx$ , $v = e^x$

I = e^x \sin x - \int e^x \cos x \, dx

Second parts for remaining integral: $u = \cos x$ , $dv = e^x dx$

$du = -\sin x \, dx$ , $v = e^x$

\int e^x \cos x \, dx = e^x \cos x + \int e^x \sin x \, dx = e^x \cos x + I

Substitute back:

I = e^x \sin x - (e^x \cos x + I) = e^x(\sin x - \cos x) - I

2I = e^x(\sin x - \cos x)

I = \frac{e^x(\sin x - \cos x)}{2} + C

Example 3: Completing the Square

Problem: $\displaystyle \int \frac{1}{x^2+4x+5} \, dx$

Solution: Complete the square:

x^2+4x+5 = (x+2)^2 + 1

Let $u = x+2$ , $du = dx$ :

\int \frac{1}{u^2+1} \, du = \tan^{-1} u + C = \tan^{-1}(x+2) + C

Example 4: Definite with Substitution

Problem: $\displaystyle \int_0^4 \frac{1}{\sqrt{x}+1} \, dx$

Solution: Let $u = \sqrt{x}$ , so $x = u^2$ and . When , ; when , .

\int_0^2 \frac{2u}{u+1} \, du = 2\int_0^2 \frac{u+1-1}{u+1} \, du = 2\int_0^2 \left(1 - \frac{1}{u+1}\right) du

= 2\left[u - \ln|u+1|\right]_0^2 = 2\left[(2-\ln 3) - (0-\ln 1)\right] = 2(2-\ln 3) = 4 - 2\ln 3

16. Practice Problems

Level 1 — Basic

$\displaystyle \int (5x^3 - 2x + 7) \, dx$
$\displaystyle \int \frac{3}{x} \, dx$

Level 2 — Substitution

$\displaystyle \int x(x^2+1)^4 \, dx$
$\displaystyle \int \frac{\cos x}{1+\sin x} \, dx$

Level 3 — By Parts & Partial Fractions

$\displaystyle \int x \ln x \, dx$
$\displaystyle \int x^2 e^x \, dx$

Level 4 — Trigonometric & Special

$\displaystyle \int \sin^2 x \cos^2 x \, dx$
$\displaystyle \int \frac{1}{\sqrt{x^2+16}} \, dx$

17. Answers to Practice Problems

Problem	Answer
1	$\frac{5x^4}{4} - x^2 + 7x + C$

Quick Reference Formula Sheet

Integral	Result
$\int x^n \, dx$	$\frac{x^{n+1}}{n+1} + C$

End of Integration Formulas Guide

Complete Integration Formulas: Basic to Intermediate

1. Basic Integration Rules

1.1 Fundamental Rules

1.2 Example: Power Rule

2. Standard Algebraic Integrals

2.1 Example: Exponential and Logarithmic

3. Trigonometric Integrals

3.1 Basic Trigonometric Formulas

3.2 Square Terms

3.3 Example: Trigonometric Integration

4. Inverse Trigonometric Integrals

4.1 Generalized Forms

4.2 Example: Inverse Trigonometric

5. Hyperbolic Integrals

6. Integration by Substitution

6.1 Method

6.2 Common Substitutions

6.3 Example 1: Linear Substitution

6.4 Example 2: Logarithmic Type

6.5 Example 3: Trigonometric Substitution

7. Integration by Parts

7.1 Formula

7.2 LIATE Rule (Priority for choosing uuu)

7.3 Example 1: Algebraic × Exponential

7.4 Example 2: Logarithmic

7.5 Example 3: Repeated Integration by Parts

8. Integration of Rational Functions (Partial Fractions)

8.1 Types of Partial Fractions

8.2 Example 1: Distinct Linear Factors

8.3 Example 2: Repeated Linear Factor

9. Trigonometric Integrals (Advanced)

9.1 Powers of Sine and Cosine

9.2 Example: Even Powers

9.3 Products to Sums

10. Trigonometric Substitution

10.1 Standard Substitutions

10.2 Example: a2−x2\sqrt{a^2-x^2}a2−x2​ Form

11. Special Algebraic Integrals

11.1 Forms with Square Roots

11.2 Example: Special Form

12. Definite Integrals

12.1 Fundamental Theorem of Calculus

12.2 Properties

12.3 Even and Odd Functions

12.4 Example: Definite Integral

13. Reduction Formulas

13.1 Wallis' Integrals

13.2 Reduction Formula for tan⁡nx\tan^n xtannx

13.3 Reduction Formula for sec⁡nx\sec^n xsecnx

14. More Standard Integrals (Quick Reference)

14.1 Tangent Half-Angle (Weierstrass Substitution)

14.2 Special Rational Forms

15. Solved Examples (Mixed Practice)

Example 1: Substitution with Root

Example 2: Integration by Parts — Cyclic

Example 3: Completing the Square

Example 4: Definite with Substitution

16. Practice Problems

Level 1 — Basic

Level 2 — Substitution

Level 3 — By Parts & Partial Fractions

Level 4 — Trigonometric & Special

17. Answers to Practice Problems

Quick Reference Formula Sheet

7.2 LIATE Rule (Priority for choosing $u$ )

10.2 Example: $\sqrt{a^2-x^2}$ Form

13.2 Reduction Formula for $\tan^n x$

13.3 Reduction Formula for $\sec^n x$