CALCULUS REVIEWER

1. Integration of Trigonometric Functions

Basic Trigonometric Integrals

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

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

\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

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Example

\int 4\sin x \, dx

Solution:

= -4\cos x + C

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2. Inverse Trigonometric Integrals

These appear when expressions look like 1/(1+x²) or 1/√(1−x²).

Important Formulas

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

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

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

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Example

\int \frac{1}{1+x^2} dx

Solution:

= \tan^{-1}x + C

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3. Integration by Parts

Used when multiplying two different types of functions.

Formula

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

Where:

u = function to differentiate

dv = function to integrate

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LIATE Rule (helps choose u)

Choose u in this order:

1. L – Logarithmic

2. I – Inverse Trig

3. A – Algebraic

4. T – Trigonometric

5. E – Exponential

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Example

\int x e^x dx

Step 1:

u = x → du = dx

dv = e^x dx → v = e^x

Step 2: Apply formula

\int x e^x dx = xe^x - \int e^x dx

= xe^x - e^x + C

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Quick Tips for Exams 📌

✔ Use basic formulas first before doing complicated methods.

✔ If you see x with e^x or sin x, think integration by parts.

✔ If you see 1/(1+x²) or 1/√(1−x²), think inverse trig integrals.

✔ Always add + C for indefinite integrals.