How to Prove Standard Integration Formulas with Examples
If a student aspires to become a successful engineer, then he or she should have a strong knowledge of integration. Integration is important in every step of engineering, from measuring cable lengths to planning long projects, integration is an integral part of Mathematics and very much important in this field of career.
There are many significant integration formulas that are used for integrating many of the standard integrals. All the different Integration methods have their own merits and play a vital role in yielding proper end results. In this section, we will look at the different integrals of the typical functions.
What is Integration? What is the Basic Formula of Integration?
Integration is generally the mixing of items that got separated earlier. If we consider the figure ∫ f(x)dx = F(x) + C, if F′(x)=f(x), ∫ is the integral symbol there. F(x) is the integrand, x is the variable, and C remains the constant of integration.
How can one Differentiate Between Definite and Indefinite Integral?
When there are integrals with a specified upper limit and lower limit, then it is referred to as a definite integral. It is also called definite because it provides a precise answer at the end of every problem that is stated. As the definite integral exists on the curve and x-axis over a specified interval, it is obtained as positive, otherwise negative when below or above the x-axis.
While when we talk about the indefinite integral, it is specified in the general terms. It is in contrast to the referred integration form. The indefinite integral is considered as functions antiderivative, as its interpretation is not possible by the state of nature. The area of an interval cannot be judged easily depending on the general nature of integration.
Check out Some Integration Formulas Below.
\[\int \frac{dx}{(x^{2} - a^{2})}\] = \[\frac{1}{2a}\] log |\[\frac{(x – a)}{(x + a)}\]| + C
\[\int \frac{dx}{(a^{2} - x^{2})}\] = \[\frac{1}{2a}\] log |\[\frac{(a - x)}{(a + x)}\]| + C
Proofs of Integration Formulas Below:
1: \[\int \frac{1}{(x^{2} - a^{2})}\] = \[\frac{1}{2a}\] log |\[\frac{(x - a)}{(x + a)}\]| + C
We know that,
\[\frac{1}{(x^{2} - a^{2})} = \frac{1}{(x - a)(x + a)} = \frac{1}{2a} [\frac{(x + a) - (x - a)}{(x - a)(x + a)}]\]
Therefore, \[\frac{1}{2a}\] [1/(x – a) – 1/(x + a)]
Therefore it goes like,
\[\int \frac{dx}{(x^{2} – a^{2})} = \frac{1}{2a}\] [ \[\int \frac{dx}{(x – a)} – \int \frac{dx}{(x + a)}\]]
= \[\frac{1}{2a}\] [log |(x – a) – log |(x + a)] + C
= \[\frac{1}{2a}\] log |(x – a) / (x + a)| + C
2: \[\int \frac{1}{(a^{2} - x^{2})}\] = \[\frac{1}{2a}\] log |\[\frac{(a - x)}{(a + x)}\]| + C
We know,
\[\frac{1}{(a^{2} - x^{2})} = \frac{1}{(a - x)(a + x)} = \frac{1}{2a} [\frac{(a + x) + (a - x)}{(a - x)(a + x)}]\]
= \[\frac{1}{2a}\] [1/(a – x) + 1/(a + x)]
Hence,
\[\int \frac{dx}{(a^{2} - x^{2})} = \frac{1}{2a}\] [ \[\int \frac{dx}{(a - x)} - \int \frac{dx}{(a + x)}\]]
= \[\frac{1}{2a}\] [– log |(a – x) + log |(a + x)] + C
= 1/2a log |(a + x) / (a – x)| + C.
After looking at the integration formulas & proof we will solve an example now.
Example 1:
Find out the integral of \[\frac{(x + 3)}{\sqrt{(5 – 4x + x^{2})}}\] with respect to x.
Solution:
We say,
W x + 3 = \[A\frac{d}{dx} \sqrt{(5 - 4x + x^{2})}\] + B = A (– 4 – 2x) + B
After equating the coefficient we get,
A = – ½ and B = 1
So, \[\frac{(x + 3)}{\sqrt{(5 – 4x + x^{2})}}\]dx = – ½ \[\int (- 4 - 2x){\sqrt{(5 - 4x + x^{2})}}\] dx + \[\int \frac{dx}{\sqrt{(5 - 4x + x^{2})}}\]
= – ½ I1 + I2 … (a)
Solving I1
We can substitute,
(5 – 4x + x2) = t,
So, (– 4 – 2x) dx = dt.
Hence
I1 = \[\int \frac{(– 4 – 2x)}{\sqrt{(5 – 4x + x^{2})}}\] dx = \[\int \frac{dt}{\sqrt{t}} = 2 \sqrt{t}\] + C1
= 2 \[\sqrt{(5 – 4x + x^{2})}\] + C1 … (b)
Solving I2
I2 = \[\frac{dx}{\sqrt{(5 – 4x + x^{2})}}\] = \[\int \frac{dx}{[9 – (x + 2)^{2}]}\]
After substituting,
(x + 2) = t,
So, dx = dt.
Hence,
I2 = \[\int \frac{dt}{\sqrt{(3^{2} – t^{2})}}\] = sin–1 (t/3) + C2
= sin–1 [(x + 2) / 3] + C2 … (c)
After substituting (b) and (c) we can get,
\[\frac{(x + 3)}{\sqrt{(5 – 4x + x^{2})}}\]dx = – ½ I1 + I2
= – \[\sqrt{(5 – 4x + x^{2})}\] + sin–1 [(x + 2) / 3] + C …
Where, C = C2 = C1/2.
After concentrating on the example let us look at some of the commonly asked questions on integration formulas.
FAQs on Understanding the Proofs of Common Integration Formulas
1. What is the proof of the power rule of integration?
The power rule of integration states that for n ≠ −1, ∫xn dx = xn+1/(n+1) + C.
The proof follows from differentiation:
- Assume F(x) = xn+1/(n+1).
- Differentiate: d/dx [xn+1/(n+1)] = xn.
- Since differentiation returns the original function, the integral must be xn+1/(n+1) + C.
2. Why is the integral of 1/x equal to ln|x| + C?
The integral of 1/x is ln|x| + C because the derivative of ln|x| is 1/x.
Proof idea:
- Let F(x) = ln|x|.
- Differentiate: d/dx [ln|x|] = 1/x for x ≠ 0.
- Since differentiation of ln|x| gives 1/x, its integral must be ln|x| + C.
3. How is the integration by parts formula proved?
The integration by parts formula is derived from the product rule of differentiation and is ∫u dv = uv − ∫v du.
Proof:
- Start with the product rule: d(uv)/dx = u dv/dx + v du/dx.
- Rewrite in differential form: d(uv) = u dv + v du.
- Integrate both sides: ∫d(uv) = ∫u dv + ∫v du.
- Simplify: uv = ∫u dv + ∫v du.
- Rearrange to get ∫u dv = uv − ∫v du.
4. How do you prove the substitution rule in integration?
The substitution rule is proved using the chain rule of differentiation and states that ∫f(g(x))g'(x) dx = ∫f(u) du.
Proof outline:
- Let u = g(x).
- Then du/dx = g'(x), so du = g'(x) dx.
- Substitute into the integral: ∫f(g(x))g'(x) dx = ∫f(u) du.
5. What is the proof of the integral of ex?
The integral of ex is ex + C because the derivative of ex is itself.
Proof:
- Let F(x) = ex.
- Differentiate: d/dx [ex] = ex.
- Since differentiation returns the same function, ∫ex dx = ex + C.
6. How is the integral of sin x proved?
The integral of sin x is −cos x + C because the derivative of −cos x equals sin x.
Proof steps:
- Let F(x) = −cos x.
- Differentiate: d/dx [−cos x] = sin x.
- Since differentiation gives sin x, the integral must be −cos x + C.
7. What is the proof of the definite integral formula using limits?
The definite integral ∫ab f(x) dx is defined as the limit of Riemann sums: limn→∞ Σ f(xi)Δx.
Proof idea:
- Divide [a, b] into n equal subintervals of width Δx.
- Form rectangles with height f(xi).
- Add areas: Σ f(xi)Δx.
- Take the limit as n → ∞.
8. Why does the constant of integration appear in indefinite integrals?
The constant of integration C appears because differentiation of any constant is zero.
Explanation:
- If F'(x) = f(x), then (F(x) + C)' = f(x).
- Different constants give the same derivative.
- Therefore, the general solution must include +C.
9. How is the integral of cos x proved?
The integral of cos x is sin x + C because the derivative of sin x equals cos x.
Proof:
- Let F(x) = sin x.
- Differentiate: d/dx [sin x] = cos x.
- Hence, ∫cos x dx = sin x + C.
10. What is the Fundamental Theorem of Calculus and how is it proved?
The Fundamental Theorem of Calculus states that if F'(x) = f(x), then ∫ab f(x) dx = F(b) − F(a).
Proof outline:
- Define F(x) = ∫ax f(t) dt.
- Using limits, show F'(x) = f(x).
- Evaluate from a to b to obtain F(b) − F(a).
