// Numbas version: exam_results_page_options {"name": "Integration 1 - Substitution (with limits)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "m"], "name": "Integration 1 - Substitution (with limits)", "tags": ["Calculus", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "Steps", "substitution"], "preamble": {"css": "", "js": ""}, "advice": "

This problem is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a[0]} * (x ^ 2) + {b[0]}}$ then $du=\\simplify[std]{(2*{a[0]} * x)*dx }$
Hence we can replace $xdx$ by $\\frac{1}{2*\\var{a[0]}}du$.

Hence the integral becomes:

\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/(2{a[0]}))u^{m[0]},u)}\\\\ &=&\\simplify[all]{(1/(2{a[0]}))u^{m[0]+1}/{m[0]+1}+C}\\\\ &=& \\simplify{({a[0]} * (x ^ 2) + {b[0]})^{m[0]+1}/(2{a[0]}*({m[0]}+1))+C} \\end{eqnarray*}\\]

If there are limits, we substitute in the upper limit for $x$ and then subtract the result of substituting in the lower limit for $x$.

This cancels the constant of integration and leaves us with a definite number.

A Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; f'(x)g(f(x))\\;dx\\]
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int g(u)\\;du \\]
And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.

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$I=\\int_{1}^{2}\\simplify[std]{x*({a[0]}x^2+{b[0]})^{m[0]} dx}$

Use $u=\\simplify[std]{{a[0]}x^2+{b[0]}}$ as your substitution.

First we integrate without the limits.

$\\frac{du}{dx}=$ [[1]]

$dx=$ [[2]]

Substituting back into the original equation for $dx$ and pulling out constants gives

$I=$[[3]]$\\int{\\simplify[std]{u^{m[0]} du}}$

The next step is to integrate.

$\\int{\\simplify{u^{m[0]} du}}=$ [[4]]

Putting all of these together, we get the result in terms of $x$ (remembering the constant of integration):

[[0]]

Now put the limits into the calculation for $x$. You may neglect $C$ in these boxes, as it will be cancelled in the subtraction.

$[$[[5]]$]-[$[[6]]$]$

$=$ [[7]]

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Input all numbers as integers or fractions and not as decimals.

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Input all numbers as integers or fractions and not as decimals.

Integrate the following by substitution.

Don't forget the constant of integration ($C$).

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Step by step solving for integration by substitution

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