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# Table of Integrals

The constant terms are omitted except where they are the point

# Basic Forms

• {$\large \int 0 dx = C$}
• {$\large \int k dx = kx$}
• {$\large \int k f(x)dx = k \int f(x) dx$}
• {$\large \int x^n dx = {{x^{n+1}} \over {n+1}} , n \ne -1$}
• {$\large \int {{dx}\over x} = \ln |x|$}

# Techniques

• {$\large \int uv' dx = uv - \int u'v dx \; \text{ (integration by parts)}$}
• {$\large \int \{f(x) + g(x)\}dx = \int f(x) dx + \int g(x)dx$}
• {$\large \int F\{f(x)\} dx= \int F(u){{dx} \over {du}}du = \int {{F(u)} \over {f^\prime(x)}}du$}

# Rational Functions

• {$\large \int {dx \over \sqrt{1 - x^2}} = \arcsin x$} (Demo)
• {$\large \int {dx \over {1 + x^2}} = \arctan x$}

# Exponential Functions

• {$\large \int f^\prime(x)e^{f(x)} dx = e^{f(x)}$}
• {$\large \int e^{cx} dx = \frac{1}{c}e^{cx}$}
• {$\large \int xe^x dx = xe^x - e^x$}
• {$\large \int x e^{x^2} dx = \frac{1}{2} e^{x^2}$}

# Trigonometric Functions

• {$\large \int \cos x dx = \sin x$}
• {$\large \int \sin x dx = -\cos x$}
• {$\large \int \tan x dx = -log \cos x$}
• {$\large \int \omega\cos\omega t dt = \sin \omega t$}
• {$\large \int \omega\sin\omega t dt = -\cos \omega t$}
• {$\large\int t \sin t dt = -t\cos t + \sin t$} (Demo)
• {$\large \int 2\sin x \cos x dx = \sin^2 x$}
• {$\large \int \cos\theta\log \sin \theta d\theta = \sin\theta( \log \sin \theta - 1)$} (Demo)
• {$\large \int \tan^2 dx = \tan x - x$}
• {$\large \int \sec^4\theta\ d\theta = \frac{1}{3}\tan^3\theta + \tan\theta$} (Demo)

# Inverse Trigonometric Functions

• {$\large\int \arcsin\ x dx = x\ \arcsin\ x + \sqrt{1 - x^2}$} (Demo)
• {$\large \int x^2\ \arcsin\ x\ dx = \frac{1}{3}x^3 \arcsin x + \frac{1}{9}x^2\sqrt{1-x^2} + \frac{2}{9}\sqrt{1-x^2}$} (Demo)

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