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 Indefinite Integral
4 Integration of a Function Raised to a Power
5 Integral of a General Polynomial
6 Constant of Integration
7 Integration of Reciprocal Powers
8 Linearity of Integration
9 Integral of an Exponential with Linear Exponent
10 Integral of a Constant
11 Integral of a General Exponential
12 Power Rule of Integration
13 Integral of a General Base Exponential
14 Integral of Reciprocal Function
15 Integral of a Linear Exponential
16 Exponential Representation of Hyperbolic Functions
17 Integration of Reciprocal Linear Function
18 Integral of a Logarithm
19 Integral of a Logarithm with Arbitrary Base
20 Integral of a Power of Logarithm
21 Integral of $$\frac{\ln x}{x}$$
22 Integration by Parts
23 Integral of $$\frac{(\ln x)^n}{x}$$
24 Inverse Trigonometric Integrals
25 Integration by Substitution
26 Integration by Parts
27 Definite Integral
28 Integral of a Power of Sine
29 Fundamental Theorem of Calculus
30 Integral of a Power of Cosine
31 Power-Reduction Identity for Sine
32 Fundamental Theorem with Variable Upper Limit
33 Fundamental Theorem with Variable Lower Limit
34 Power-Reduction Identity for Cosine
35 Integral with Both Limits as Functions
36 Double-Angle Identity for Sine
37 Additivity of Definite Integrals
38 Integral of $$\tan^2x$$
39 Reduction Formula for Logarithmic Powers
40 Integral of $$\sec^3x$$
41 Double-Angle Identity for Cosine
42 Reversal of Limits
43 Zero-Length Integral
44 Linearity of Definite Integration
45 Integral of a Power Function
46 Integral of Reciprocal Function over a Positive Interval
47 Integral of $$\tan x\sec x$$
48 Even Function Property
49 Integral of $$\cot x\csc x$$
50 Alternative Double-Angle Forms
51 Integral of $$\frac{1}{\sqrt{1-x^2}}$$
52 Symmetry About a Point
53 General Inverse-Tangent Form
54 Comparison Property
55 Trigonometric Integrals
56 Integral of $$\sin x\cos x$$
57 Odd Function Property
58 Integral of $$\frac{1}{1+x^2}$$
59 Integration of a Polynomial Times Exponential
60 Improper Rational Function
61 Non-Negative Integral
62 Integral of a Reciprocal Square-Root Sum
63 Average Value of a Function
64 Integral of $$\frac{1}{x\sqrt{x^2+a^2}}$$
65 Area Under a Curve
66 Integral of $$\frac{1}{x\sqrt{x^2-a^2}}$$
67 Area Between Two Curves
68 Integration of an Even Function over a Symmetric Interval
69 Additivity Over Adjacent Intervals
70 Symmetry Under $$x\rightarrow a-x$$
71 Displacement from Velocity
72 Velocity from Acceleration
73 Integral of a Periodic Function Over a Symmetric Period
74 Total Distance from Velocity
75 Integration by Substitution
76 Area in Parametric Form
77 Area Enclosed by a Curve and the Coordinate Axes
78 Area in Polar Coordinates
79 Trigonometric Identities for Integration
80 Mean Value Theorem for Integrals
81 Arc Length in Cartesian Form
82 Arc Length in Parametric Form
83 Integral of an Odd Power
84 Volume by Revolution About the x-Axis
85 Special Exponential Integrals
86 Substitution $$x=a-t$$
87 Volume by Revolution About the y-Axis
88 Basic Integration Formulas
89 Volume About an Arbitrary Horizontal Line
90 Volume Using Horizontal Washers
91 Standard Integral of a Difference of Squares
92 Standard Form for Difference of Squares in Variable Shift
93 Washer Method
94 Shell Method
95 Integral of $$\frac{x}{x^2+a^2}$$
96 General Cross-Section Volume
97 Disc Method
98 Integral of a Product of Different-Frequency Cosines
99 Volume About an Arbitrary Vertical Line
100 Beta-Gamma Relation
101 Gamma Function
102 Reduction Formula for Exponential-Power Integral
103 Area Between Intersecting Curves
104 Integration of Higher Powers of a Linear Denominator
105 Wallis Integral for Even Powers
106 Substitution in Definite Integrals with Reversed Limits
107 Integral of $$\frac{1}{a^2-x^2}$$
108 Frullani's Integral
109 Differential in Weierstrass Substitution
110 Integral of Hyperbolic Cosecant Squared
111 Cover-Up Method
112 Comparison Near Zero
113 Integral of a Nonnegative Function
114 Integral of a Product of Nonnegative Functions
115 Secant and Cosecant Powers
116 Orthogonality of Different Cosine Frequencies
117 Minkowski Integral Inequality
118 Area Between a Curve and Its Reflection
119 Area Between a Curve and the x-Axis
120 Standard Integral of Cotangent
121 Wallis Product
122 Improper Integral Over the Entire Real Line
123 Comparison Test for Improper Integrals
124 Reciprocal Substitution
125 Leibniz Rule
126 Average Value of a Function
127 Homogeneous Substitution
128 Translation of Variable
129 Integral of a Derivative
130 Integral of a Polynomial on a Symmetric Interval
131 Definite Integral with Linear Substitution
132 Surface Area of Revolution About the y-Axi
133 Improper Integral with Infinite Lower Limit
134 Integration of a General Power
135 Improper Integral with an Infinite Discontinuity
136 Integral of Derivative over Function
137 Reversal of Limits
138 Integral with Equal Limits
139 Repeated Irreducible Quadratic
140 Trigonometric Substitution for $$\sqrt{a^2+x^2}$$
141 Integral of a Shifted Difference of Squares
142 Integration of a Rational Function
143 Completing the Square for Positive Constant Term
144 Dirichlet-Type Integral
145 Wallis Integral for Odd Powers
146 Integral of $$\frac{2x+b}{x^2+bx+c}$$
147 Mean Value of a Continuous Function
148 Integral of Inverse Cosine
149 Improper Integral with an Interior Singularity
150 Euler Substitution for Positive Leading Coefficient
151 Trigonometric Beta-Gamma
152 Differentiation of an Integral
153 Second Derivative of an Integral
154 Integral Equation
155 Second Fundamental Theorem of Calculus
156 Exponential and Logarithmic Forms
157 Differentiating an Integral Equation
158 Area as an Integral
159 Integral of $$\cot^2x$$
160 Combining Reciprocal Forms
161 Reciprocal Polynomial Transformation
162 Beta Function
163 Volume of a Solid with Square Cross-Sections
164 Volume of a Solid with Equilateral-Triangle Cross-Sections
165 Logarithmic Case of the Power Rule
166 Linearity of Indefinite Integration
167 Linearity of Definite Integration
168 Integral of a Quadratic with Distinct Real Roots
169 Rationalizing Substitution
170 Integral of an Exact Differential
171 Integral Absolute-Value Inequality
172 Sine Integration Pattern
173 Comparison Property
174 Bound Using Minimum and Maximum Values
175 Integrals of Form ∫ dx/(a sin x + b cos x)
176 Rational Functions of Sine/Cosine
177 Integral Mean Value Theorem
178 Area Between a Curve and a Line
179 Secant-Tangent Integration Pattern
180 Tangent Integration Pattern
181 Polar Arc Length Differential
182 Cotangent Integration Pattern
183 Polar Coordinate Conversion
184 Integral of $$\frac{1}{x^2-a^2}$$ by Partial Fractions
185 Area Using Horizontal Strips
186 Double Factorial
187 Wallis Reduction Formula
188 Symmetry Weighted Integral for Two Functions
189 Reciprocal Symmetry on $$[0,\infty)$$
190 Scaling Property of Integrals
191 Trigonometric Identity for Tangent
192 Standard Forms with Quadratic Denominators
193 General Surface of Revolution
194 Integral of a Function with Absolute Value
195 Trigonometric Substitution for $$\sqrt{x^2-a^2}$$
196 Integral of a Quadratic Function
197 Cartesian Distance in Polar Coordinates
198 Integral of a Product of Different-Frequency Sines
199 LIATE Rule
200 Repeated Integration by Parts
201 Integration by Parts Selection Rule
202 Comparison Test for Divergence
203 Integral of a Shifted Radical
204 Equality Condition for Absolute-Value Inequality
205 Chebyshev Integral Inequality
206 Integral of Cosine Squared Over a Complete Period
207 Integral of a Constant over an Interval
208 Integration of a Composite Power
209 Integral of a Quadratic Derivative Pattern
210 Integral of Exponential-Trigonometric Product
211 Integration by Parts for Definite Integrals
212 Parameter Differentiation Under the Integral Sign
213 Trigonometric Substitution for $$\sqrt{a^2-x^2}$$
214 Antiderivative
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