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<h1>Substitution</h1><p>The fundamental theorem relates the derivative and an antiderivative via:</p>$$~
\int_a^b f(x) dx = F(b) - F(a).
~$$<p>If $F(x) = G(u(x))$, then the chain rule tells us that $f(x) = g(u(x)) u'(x)$, so we get:</p>$$~
\int_{u(a)}^{u(b)} g(x) dx = \int_a^b g(u(x)) u'(x) dx= G(u(x)) \big|_a^b = G(x) \big|_{u(a)}^{u(b)}.
~$$<p>That is, this substitution formula applies:</p><blockquote>
$$ \int_a^b g(u(x)) u'(x) dx = \int_{u(a)}^{u(b)} g(x) dx. $$
</blockquote><p>We've seen a special case of this where $u(x) = x-c$ in the formula $\int_{a-c}^{b-c} g(x) dx= \int_a^b g(x-c)dx$.</p><p>The main use of this is to take complicated things inside of the function $g$ out of the function (the $u(x)$) then account for the change.</p><p>Some examples are in order.</p><p>Consider:</p>$$~
\int_0^{\pi/2} \cos(x) e^{\sin(x)} dx.
~$$<p>Clearly the $\sin(x)$ inside the exponential is an issue. If we let $u(x) = \sin(x)$, then $u'(x) = \cos(x)$, and this becomes</p>$$~
\int_{u(0)}^{u(\pi/2)} e^x dx = e^x \big|_{\sin(0)}^{\sin(\pi/2)} = e^1 - e^0.
~$$<p>This all worked as the problem was such that it was more or less obvious what to choose for $u$ and $G$.</p><h3>"u" substitution</h3><p>The process of identifying the result of the chain rule in the function to integrate is not automatic, but rather a bit of an art. The basic step is to try some values. Typically, this is taught by "substituting" in some value for part of the expression (basically the $u(x)$) and seeing what happens.</p><p>In the above problem, $\int_0^{\pi/2} \cos(x) e^{\sin(x)} dx$, we might just rename $\sin(x)$ to be $u$ (suppressing the "of $x$ part). Then we need to rewrite the "$dx$" part of the integral. We know in this case that $du/dx = \cos(x)$. In terms of differentials, this gives $du = \cos(x) dx$. But this allows us to substitute in with $u$ and $du$ as is possible:</p>$$~
\int_0^{\pi/2} \cos(x) e^{\sin(x)} dx = \int_0^{\pi/2} e^{\sin(x)} \cdot \cos(x) dx = \int_{u(0)}^{u(\pi)} e^u du.
~$$<hr /><p>Let's illustrate with a new problem: $\int_0^2 4x e^{x^2} dx$.</p><p>Again, we see that the $x^2$ inside the exponential is a complication. Letting $u = x^2$ we have $du = 2x dx$. We have $4xdx$ in the original problem, so we will end up with $2du$:</p>$$~
\int_0^2 4x e^{x^2} dx = 2\int_0^2 e^{x^2} \cdot 2x dx = 2\int_{u(0)}^{u(2)} e^u du = 2 \int_0^4 e^u du =
2 e^u\big|_{u=0}^4 = 2(e^4 - 1).
~$$<hr /><p>Consider now $\int_0^1 2x^2 \sqrt{1 + x^3} dx$. Here we see that the $1 + x^3$ makes the square root term complicated. If we call this $u$, the what is $du$? Clearly, $du = 3x^2 dx$, or $(1/3)du = x^2 dx$, so we can rewrite this as:</p>$$~
\int_0^1 2x^2 \sqrt{1 + x^3} dx = \int_{u(0)}^{u(1)} 2 \sqrt{u} (1/3) du = 2/3 \cdot \frac{u^{3/2}}{3/2} \big|_1^2 =
\frac{4}{9} \cdot(2^{3/2} - 1).
~$$<hr /><p>Consider $\int_0^{\pi} \cos(x)^3 \sin(x) dx$. The $\cos(x)$ function inside the $x^3$ function is complicated. We let $u(x) = \cos(x)$ and see what that implies: $du = \sin(x) dx$, which we see is part of the question. So the above becomes:</p>$$~
\int_0^{\pi} \cos(x)^3 \sin(x) dx = \int_{u(0)}^{u(\pi)} u^3 du= \frac{u^4}{4}\big|_0^0 = 0.
~$$<p>Changing limits leaves the two endpoints the same, which means the total area after substitution is $0$. A graph of this function shows that around $\pi/2$ we can see the function has symmetry about $\pi/2$, so the answer of $0$ is supported by the plot:</p><pre class="sourceCode julia">using Plots
f(x) = cos(x)^3 * sin(x)
plot(f, 0, 1pi)</pre>
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// AMD support
if (typeof define === "function" && define.amd) {
// Define as an anonymous module
define("Gadfly", ["Snap.svg"], function (Snap) {
return factory(Snap);
});
} else {
// Browser globals (glob is window)
// Snap adds itself to window
glob.Gadfly = factory(glob.Snap);
}
}(this, function (Snap) {
var Gadfly = {};
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var xPX = function(fig, x) {
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var yPX = function(fig, y) {
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};
Snap.plugin(function (Snap, Element, Paper, global) {
// Traverse upwards from a snap element to find and return the first
// note with the "plotroot" class.
Element.prototype.plotroot = function () {
var element = this;
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element = element.parent();
}
return element;
};
Element.prototype.svgroot = function () {
var element = this;
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element = element.parent();
}
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};
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x1: bbox.x + bbox.width,
y0: bbox.y,
y1: bbox.y + bbox.height
};
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};
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// See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/
var events = ["mouseenter", "mouseleave"];
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(function (event_name) {
var event_name = events[i];
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return;
}
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}
};
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this.mouseover(fn2, scope);
} else {
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}
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fn.apply(el, [event]);
};
}
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};
// Snap's attr function can be too slow for things like panning/zooming.
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}
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this.mouseenter(Gadfly.plot_mouseover)
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Gadfly.guide_background_drag_onstart,
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this.mouseenter(function (event) {
init_pan_zoom(this.plotroot());
});
return this;
};
});
// When the plot is moused over, emphasize the grid lines.
Gadfly.plot_mouseover = function(event) {
var root = this.plotroot();
var keyboard_zoom = function(event) {
if (event.which == 187) { // plus
increase_zoom_by_position(root, 0.1, true);
} else if (event.which == 189) { // minus
increase_zoom_by_position(root, -0.1, true);
}
};
root.data("keyboard_zoom", keyboard_zoom);
window.addEventListener("keyup", keyboard_zoom);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
xgridlines.data("unfocused_strokedash",
xgridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
ygridlines.data("unfocused_strokedash",
ygridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
// emphasize grid lines
var destcolor = root.data("focused_xgrid_color");
xgridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
destcolor = root.data("focused_ygrid_color");
ygridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);
// reveal zoom slider
root.select(".zoomslider")
.animate({opacity: 1.0}, 250);
};
// Reset pan and zoom on double click
Gadfly.plot_dblclick = function(event) {
set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0);
};
// Unemphasize grid lines on mouse out.
Gadfly.plot_mouseout = function(event) {
var root = this.plotroot();
window.removeEventListener("keyup", root.data("keyboard_zoom"));
root.data("keyboard_zoom", undefined);
var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");
var destcolor = root.data("unfocused_xgrid_color");