Indefinite Integration

Formulas are organized based on their importance in the CAT exam.
1 Integral of a Polynomial Times a Linear Factor
2 Partial Fractions
3 Constant of Integration
4 Power Rule of Integration
5 Integration of Reciprocal Powers
6 Integrals of Form ∫ dx/(a sin x + b cos x)
7 Integral of a Constant
8 Non-Negative Integral
9 Exponential Representation of Hyperbolic Functions
10 Integral of a Logarithm
11 Integral of a General Polynomial
12 Integral of a General Exponential
13 Integral of $$\frac{(\ln x)^n}{x}$$
14 Even Function Property
15 Comparison Property
16 Integral of a Linear Exponential
17 Integration by Parts
18 Power-Reduction Identity for Sine
19 Fundamental Theorem with Variable Upper Limit
20 Power-Reduction Identity for Cosine
21 Fundamental Theorem with Variable Lower Limit
22 Integral of $$\frac{\ln x}{x}$$
23 Linearity of Integration
24 Odd Function Property
25 Equality Condition for Absolute-Value Inequality
26 Fundamental Theorem of Calculus
27 Integration of a Function Raised to a Power
28 Area in Polar Coordinates
29 Integral of $$\sec^3x$$
30 Reversal of Limits
31 Double-Angle Identity for Cosine
32 Arc Length in Parametric Form
33 Substitution $$x=a-t$$
34 Integral of $$\cot x\csc x$$
35 Integral of $$\frac{1}{\sqrt{1-x^2}}$$
36 Disc Method
37 Integral of a Product of Different-Frequency Cosines
38 Integration of Reciprocal Linear Function
39 Inverse Trigonometric Integrals
40 Integration by Parts
41 Integration by Substitution
42 Additivity of Definite Integrals
43 Integral of Reciprocal Function over a Positive Interval
44 Integration by Parts for Definite Integrals
45 Integral of $$\tan x\sec x$$
46 Integral of $$\frac{1}{x\sqrt{x^2+a^2}}$$
47 Definite Integral
48 Indefinite Integral
49 Reduction Formula for Logarithmic Powers
50 Integral of $$\frac{1}{1+x^2}$$
51 Integral of a Power of Sine
52 Double-Angle Identity for Sine
53 Logarithmic Case of the Power Rule
54 Alternative Double-Angle Forms
55 Area Between Intersecting Curves
56 Reduction Formula for Exponential-Power Integral
57 Integral of a Power of Cosine
58 Trigonometric Integrals
59 Zero-Length Integral
60 General Inverse-Tangent Form
61 Integral of a Power Function
62 Integration of a Polynomial Times Exponential
63 Integral of a Quadratic Derivative Pattern
64 Integral of a Power of Logarithm
65 Integral of a General Base Exponential
66 Velocity from Acceleration
67 Integral of a Periodic Function Over a Symmetric Period
68 Total Distance from Velocity
69 Integration of an Even Function over a Symmetric Interval
70 Displacement from Velocity
71 Integral of $$\tan^2x$$
72 Symmetry Under $$x\rightarrow a-x$$
73 Integral of a Logarithm with Arbitrary Base
74 Integration by Substitution
75 Area Enclosed by a Curve and the Coordinate Axes
76 Area in Parametric Form
77 Integral of a Reciprocal Square-Root Sum
78 Integral of $$\sin x\cos x$$
79 Linearity of Definite Integration
80 Improper Rational Function
81 Integral with Both Limits as Functions
82 Trigonometric Identities for Integration
83 Additivity Over Adjacent Intervals
84 Integral of $$\frac{1}{x\sqrt{x^2-a^2}}$$
85 Area Under a Curve
86 Average Value of a Function
87 Integral of an Exponential with Linear Exponent
88 Mean Value Theorem for Integrals
89 Area Between Two Curves
90 Symmetry About a Point
91 Volume by Revolution About the x-Axis
92 Integral of Reciprocal Function
93 Arc Length in Cartesian Form
94 Special Exponential Integrals
95 Integral of an Odd Power
96 Basic Integration Formulas
97 Volume About an Arbitrary Horizontal Line
98 Volume by Revolution About the y-Axis
99 Standard Form for Difference of Squares in Variable Shift
100 Integral of $$\frac{x}{x^2+a^2}$$
101 General Cross-Section Volume
102 Washer Method
103 Shell Method
104 Volume About an Arbitrary Vertical Line
105 Beta-Gamma Relation
106 Gamma Function
107 Integration of Higher Powers of a Linear Denominator
108 Wallis Integral for Even Powers
109 Substitution in Definite Integrals with Reversed Limits
110 Integral of $$\frac{1}{a^2-x^2}$$
111 Standard Integral of a Difference of Squares
112 Cover-Up Method
113 Integral of Hyperbolic Cosecant Squared
114 Volume Using Horizontal Washers
115 Frullani's Integral
116 Differential in Weierstrass Substitution
117 Orthogonality of Different Cosine Frequencies
118 Area Between a Curve and the x-Axis
119 Secant and Cosecant Powers
120 Integral of a Product of Nonnegative Functions
121 Area Between a Curve and Its Reflection
122 Standard Integral of Cotangent
123 Comparison Near Zero
124 Integral of a Nonnegative Function
125 Wallis Product
126 Comparison Test for Improper Integrals
127 Minkowski Integral Inequality
128 Reciprocal Substitution
129 Leibniz Rule
130 Average Value of a Function
131 Translation of Variable
132 Integral of a Derivative
133 Integral of a Polynomial on a Symmetric Interval
134 Definite Integral with Linear Substitution
135 Integration of a General Power
136 Improper Integral with an Infinite Discontinuity
137 Improper Integral Over the Entire Real Line
138 Homogeneous Substitution
139 Integral with Equal Limits
140 Integral of Derivative over Function
141 Repeated Irreducible Quadratic
142 Reversal of Limits
143 Trigonometric Substitution for $$\sqrt{a^2+x^2}$$
144 Integral of a Shifted Difference of Squares
145 Integration of a Rational Function
146 Dirichlet-Type Integral
147 Wallis Integral for Odd Powers
148 Integral of $$\frac{2x+b}{x^2+bx+c}$$
149 Mean Value of a Continuous Function
150 Integral of Inverse Cosine
151 Euler Substitution for Positive Leading Coefficient
152 Surface Area of Revolution About the y-Axi
153 Improper Integral with Infinite Lower Limit
154 Second Derivative of an Integral
155 Second Fundamental Theorem of Calculus
156 Exponential and Logarithmic Forms
157 Completing the Square for Positive Constant Term
158 Area as an Integral
159 Integral of $$\cot^2x$$
160 Trigonometric Beta-Gamma
161 Differentiation of an Integral
162 Integral Equation
163 Beta Function
164 Volume of a Solid with Square Cross-Sections
165 Volume of a Solid with Equilateral-Triangle Cross-Sections
166 Reciprocal Polynomial Transformation
167 Linearity of Definite Integration
168 Differentiating an Integral Equation
169 Integral Absolute-Value Inequality
170 Improper Integral with an Interior Singularity
171 Combining Reciprocal Forms
172 Bound Using Minimum and Maximum Values
173 Linearity of Indefinite Integration
174 Secant-Tangent Integration Pattern
175 Integral of a Quadratic with Distinct Real Roots
176 Rationalizing Substitution
177 Integral of an Exact Differential
178 Comparison Property
179 Sine Integration Pattern
180 Polar Arc Length Differential
181 Cotangent Integration Pattern
182 Rational Functions of Sine/Cosine
183 Tangent Integration Pattern
184 Polar Coordinate Conversion
185 Integral of $$\frac{1}{x^2-a^2}$$ by Partial Fractions
186 Area Using Horizontal Strips
187 Wallis Reduction Formula
188 Symmetry Weighted Integral for Two Functions
189 Reciprocal Symmetry on $$[0,\infty)$$
190 Scaling Property of Integrals
191 Double Factorial
192 Trigonometric Identity for Tangent
193 Standard Forms with Quadratic Denominators
194 General Surface of Revolution
195 Integral of a Function with Absolute Value
196 Trigonometric Substitution for $$\sqrt{x^2-a^2}$$
197 Integral of a Quadratic Function
198 Cartesian Distance in Polar Coordinates
199 Integral of a Product of Different-Frequency Sines
200 LIATE Rule
201 Repeated Integration by Parts
202 Comparison Test for Divergence
203 Integral of a Shifted Radical
204 Chebyshev Integral Inequality
205 Integral of Cosine Squared Over a Complete Period
206 Integral of a Constant over an Interval
207 Integration of a Composite Power
208 Integral Mean Value Theorem
209 Area Between a Curve and a Line
210 Integration by Parts Selection Rule
211 Parameter Differentiation Under the Integral Sign
212 Trigonometric Substitution for $$\sqrt{a^2-x^2}$$
213 Antiderivative
214 Integral of Exponential-Trigonometric Product
215 Standard Integral of Secant
216 Integral of a Rational Function After Polynomial Division
217 Integration of a Function of $$ax+b$$
218 Surface Area of Revolution About the x-Axis
219 Partial Fraction for Repeated Linear Factors
220  Integral of a Linear Function Times Its Logarithm
221 Integral of Logarithm with Linear Argument
222 Integral of Reciprocal Quadratic with Negative Discriminant
223 Integral of a Derivative
224 Standard Integral with Difference Radical
225 Gamma-Factorial Relation
226 Integration of 1/(sin^m x cos^n x)
227 First Fundamental Theorem of Calculus
228 Chain Rule Recognition
229 Substitution in an Integral with Reciprocal Terms
230 Area Between Two Curves
231 Area Enclosed by a Closed Parametric Curve
232 Area Between Symmetric Curves
233 Reduction Formula for Sine Powers
234 Polar Area Element
235 Polar Area Between Two Curves
236 Both Limits Variable
237 Integral of a Product with a Symmetric Function
238 Area Between a Curve and the y-Axis
239 Volume of a Solid with Circular Cross-Sections
240 Standard Integral of a Reciprocal Quadratic
241 Arc Length Differential
242 Integration of a Reciprocal Linear Function
243 Cyclic Integration by Parts
244 Integration of Inverse Trigonometric Functions
245 Integral of Exponential-Sine Product
246 Reduction Formula for Cosecant Powers
247 Integral Mean Value Theorem
248 Symmetry Weighted Integral for a Linear Function
249 Translation Invariance Over Equal Intervals
250 Integral of a Shifted Sum of Squares
251 Telescoping Integral
252 Integral of Inverse Cotangent
253 Hyperbolic Tangent Identity
254 Polar Area Between Angles
255 Trigonometric Identity for Cotangent
256 Integral of $$\frac{1}{x^2-a^2}$$
257 Arc Length of a Circle
258 Integral of a Quadratic Reciprocal
259 Integration of a Constant
260 Volume of a Solid with Semicircular Cross-Sections
261 Completing the Square
262 Equivalent Cosecant Integral Forms
263 Integral of Secant Using Rationalization
264 Partial Fraction for Distinct Linear Factors
265 Completing the Square for a Quadratic
266 Standard Form for Sum of Squares
267 Integral of a Linear Function
268 Integral of Cotangent Using Logarithm
269 Dirichlet Kernel Integral
270 Position from Acceleration
271 Integral of Absolute Value
272 Integral of $$\frac{1}{a^2-x^2}$$ by Partial Fractions
273 Integral of Cosine Over a Complete Period
274 Total Geometric Area
275 Volume by Cross-Sections
276 Integral of a Function with Period $$t$$ Over Half-Shifted Intervals
277 Arc Length in Polar Form
278 Hyperbolic Identity
279 Integral of a Periodic Function with Zero Mean
280 Integral of Hyperbolic Secant Squared
281 Green's Theorem Area Formula
282 Euler Substitution for a Quadratic Radical
283 Wallis Integral
284 Area Under a Parametric Curve
285 Integral of Sine Over a Complete Period
286 Standard Hyperbolic Substitution
287 Integral of Sine-Cosine Product Over a Complete Period
288 Position from Velocity
289 Integral of a Reciprocal Sum of Squares
290 Partial Fraction for Irreducible Quadratic
291 Integration of Rational Function with Linear Denominator
292 General Symmetry Weighted Integral
293 Symmetric Product Identity
294 Integral Bound
295 Exact Differential Pattern
296 Cosecant-Cotangent Integration Pattern
297 Signed Area
298 Area Between a Curve and the x-Axis
299 Integration of a Linear Power
300 Reduction Formula for Secant Powers
301 Integral of Cosecant Using Rationalization
302 Area Enclosed by a Curve and the y-Axis
303 Pairing Symmetric Integrands
304 Symmetric Difference Identity
305 Integral Over Multiple Periods
306 Area of a Polar Curve
307 Standard Integral of Secant-Tangent
308 Standard Integral of Tangent
309 Standard Integral of a Sum of Squares with Linear Numerator
310 Cauchy-Schwarz Integral Inequality
311 Coefficient Comparison in Partial Fractions
312 Parametric Surface Area About the y-Axis
313 Work Done by a Variable Force
314 Area of a Region Symmetric About the y-Axis
315 Periodic Function with Zero Mean
316 Accumulation Function
317 Integration of a Function of a Quadratic
318 Integration of a Function with Its Derivative
319 Logarithmic Integration Pattern
320 Exponential Integration Pattern
321 Reciprocal Symmetry on $$[a,b]$$
322 Quotient Trigonometric Identities
323 Cosine Integration Pattern
324 Reflection Property
325 General Inverse-Sine Form
326 Standard Integral of Cosine
327 Standard Integral of Cosecant Squared
328 Standard Integral of Cosecant-Cotangent
329 Improper Integral with Infinite Upper Limit
330 Standard Integral of Cosecant
331 Integral of a Reciprocal Square-Root Difference
332 Reduction Formula for Inverse Tangent Powers
333 Standard Integral with Linear Denominator and Quadratic Radical
334 Reduction Formula
335 Product of Sine and Cosine Powers
336 Tangent-Secant Power Integral
337 Partial Fraction Decomposition
338 Product-to-Sum Identity for Sine-Cosine
339 Integral of Sine-Cosine Product
340 Integral of a Reciprocal Difference of Squares
341 Integration of Rational Functions with Quadratic Denominator
342 Mass from Variable Linear Density
343 Product-to-Sum Identity for Cosine
344 Integral of a Square Root of a Quadratic
345 Standard Form for Difference of Squares
346 Substitution with Symmetric Limits
347 Reduction Formula for Cosine Powers
348 Periodic Function Integral
349 Integration of a Symmetric Function with a Linear Weight
350 p-Integral
351 Standard Integral of Sine
352 Integration of $$\sqrt{a^2-x^2}$$
353 Integral of Inverse Secant
354 Reciprocal Trigonometric Identities
355 Integral of Tangent Using Logarithm
356 Area of a Circular Sector
357 Integral of a Quadratic Polynomial
358 Integral of $$\csc^3x$$
359 Periodic Shift Property
360 Parametric Surface Area About the x-Axis
361 Endpoint Evaluation Notation
362 Integral of an Even Power on a Symmetric Interval
363 Integral of $$\frac{1}{a^2+x^2}$$
364 Integrals of Powers of Tangent and Cotangent
365 Splitting an Integral at Zeros
366 Variable Upper Limit
367 Integration of $$\sqrt{x^2+a^2}$$
368 Partial Fractions with Irreducible Quadratic Factors
369 Polar Surface Area About the Polar Axis
370 Same-Frequency Sine Integral
371 Integral of Inverse Tangent
372 Integral of Inverse Cosecant
373 Reverse Chain Rule
374 Area of a Region Symmetric About the x-Axis
375 Integral of Reciprocal Quadratic with Positive Discriminant
376 Standard Integral of Secant Squared
377 Integral of Hyperbolic Cosine
378 Weighted Average Using Integration
379 Change of Variable Formula
380 Beta Integral in Trigonometric Form
381 Integral of Reciprocal Quadratic with Zero Discriminant
382 Weierstrass Substitution
383 Orthogonality of Sine and Cosine
384 Standard Integral of a Quadratic Radical
385 Partial Fractions with Repeated Linear Factors
386 Standard Integral of a Quadratic Denominator
387 Integration of a Composite Linear Argument
388 Completing the Square for Negative Constant Term
389 Orthogonality of Different Sine Frequencies
390 Integration of a Polynomial Times Trigonometric Function
391 Integration of Polynomial Times Logarithm
392 Integration of a Periodic Function Over an Integer Number of Periods
393 Area Using Horizontal Slices
394 Product-to-Sum Identity for Sine
395 Integral of an Even Power
396 General Area Between Two Curves
397 Integration of an Odd Function over a Symmetric Interval
398 Integration of $$\sqrt{x^2-a^2}$$
399 Product of Sine and Cosine Powers with Even Powers
400 Area Between Two Polar Curves
401 Integral of Hyperbolic Sine
402 Integration of a Perfect-Square Denominato
403 Half-Angle Substitution
404 Variable Lower Limit
405 Integral of Sine Squared Over a Complete Period
406 Polar Surface Area About the Initial Line
407 Cotangent-Cosecant Power Integral
408 Integration by Substitution in Definite Integrals
409 Tangent-Secant Integral for Odd Secant Power
410 Integral of $$\frac{x}{x^2-a^2}$$
411 Integration as the Inverse of Differentiation
412 Same-Frequency Cosine Integral