 # Trigonometric integration

In engineering, rotational action is often translated into lineal action, or vice-versa. Sine, cosine, and tangent, are three trigonometric functions which describe lateral and transverse displacements, and their ratio, as a radius rotates through the circle it describes.  ## Summary of Integral Properties and Solutions

\$∫sin x dx = -cos x + C\$

\$∫cos x dx = sin x + C\$

\$∫sin (ax + b) dx = -1/{a}cos (ax + b) + C\$

\$∫cos (ax + b) dx = 1/{a}sin(ax + b) + C\$

### Trigonometric substitutions

If the integrand contains a quadratic radical expression, these trig substitutions may be used:

\$√(a^2 - x^2)\$ ⇒ \$x = a⋅sin(θ)\$

\$√(x^2 - a^2)\$ ⇒ \$x = a⋅sec(θ)\$

\$√(x^2 + a^2)\$ ⇒ \$x = a⋅tan(θ)\$

f(x)F(x)f(x)F(x)
\$a\$\$ax\$\$x^n\$\${x^{n+1}}/{n+1}\$
\$1/x\$ln|\$x\$|\$1/{x^n}\$\${-1}/{(n-1)x^{n-1}}\$
\$√x\$\${2/3}x√x\$\$1/{√x}\$\$2√x\$
\$1/{x^2+a^2}\$\${1/a}\$arctan\$(x/a)\$\$1/{x^2-a^2}\$\$1/{2a}\$ln\$|{x-a}/{x+b}|\$
\$e^x\$\$e^x\$ln\$(x)\$\$x(\$ln\$(x)-1)\$
\$a^x\$\${a^x}/{ln(a)}\$log\$_a(x)\$\$x(\$log\$_a(x)-\$log\$_a(e))\$
\$xe^{ax}\$\$1/{a^2}(ax-1)e^{ax}\$\$x\$ln\$(ax)\$\${x^2}/4(2\$ln\$(ax)-1)\$
sin\$(x)\$-cos\$(x)\$arcsin\$(x)\$\$x\$arcsin\$(x)+√{1-x^2}\$
cos\$(x)\$sin\$(x)\$arccos\$(x)\$\$x\$arccos\$(x)-√{1-x^2}\$
tan\$(x)\$-ln|cos\$(x)\$|arctan\$(x)\$\$x\$arctan\$(x)-1/2\$ln\$(1+x^2)\$
cot\$(x)\$ln|sin\$(x)\$|arccot\$(x)\$\$x\$arccot\$(x)+1/2\$ln\$(1+x^2)\$
sin\$^2(x)\$\$1/2(x-\$sin\$(x)\$cos\$(x))\$\$1/{sin^2(x)}\$-cot\$(x)\$
cos\$^2(x)\$\$1/2(x+\$sin\$(x)\$cos\$(x))\$\$1/{cos^2(x)}\$tan\$(x)\$
tan\$^2(x)\$tan\$(x)-x\$\$1/{sin(x)}\$ln\$|{1-cos(x)}/{sin(x)}|\$
cot\$^2(x)\$-cot\$(x)-x\$\$1/{cos(x)}\$ln\$|{1+sin(x)}/{cos(x)}|\$
\$1/{1+sin(x)}\$\${-cos(x)}/{1+sin(x)}\$\$1/{1-sin(x)}\$\${cos(x)}/{1-sin(x)}\$
\$1/{1+cos(x)}\$\${sin(x)}/{1+cos(x)}\$\$1/{1-cos(x)}\$\${-sin(x)}/{1-cos(x)}\$
\$x\$sin\$(ax)\$\$-{1/a}x\$cos\$(ax)+1/{a^2}\$sin\$(ax)\$\$x\$cos\$(ax)\$\${1/a}x\$sin\$(ax)+1/{a^2}\$cos\$(ax)\$
\$e^{ax}\$sin\$(bx)\$\${e^{ax}}/{a^2+b^2}(a\$sin\$(bx)-b\$cos\$(bx))\$\$e^{ax}\$cos\$(bx)\$\${e^{ax}}/{a^2+b^2}(a\$cos\$(bx)+b\$sin\$(bx))\$
sinh\$(x)\$cosh\$(x)\$arsinh\$(x)\$\$x\$arsinh\$(x) - √{x^2+1}\$
cosh\$(x)\$sinh\$(x)\$arcosh\$(x)\$\$x\$arcosh\$(x) - √{x^2-1}\$
tanh\$(x)\$ln(cosh\$(x)\$)artanh\$(x)\$\$x\$artanh\$(x) +1/2\$ln\$(1-x^2)\$
coth\$(x)\$ln|sinh\$(x)\$|arcoth\$(x)\$\$x\$arcoth\$(x) +1/2\$ln\$(x^2-1)\$
\$√{x^2+a}\$\$1/2x√{x^2+a} + a/2\$ln\$|x+√{x^2+a}|\$\$1/{√{x^2+a}}\$ln\$|x+√{x^2+a}|\$
\$√{r^2-x^2}\$\$1/2x√{x^2-x^2} + {r^2}/2\$arcsin\$(x/r)\$\$1/{√{r^2-x^2}}\$arcsin\$(x/r)\$

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