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Differentiation and integration are crucial concepts in Calculus, typically used to learn about change. Besides Mathematics, Calculus also has applications in various scientific and economic fields, as well as in stock market analysis. Differentiation and integration formulas are complementary to each other, as the result of the differentiation of a function gives back the original function after integration. The original function is provided as a result of integrating the derivative of a function. Integral calculus is also known as the antiderivative because integration is the process of differentiation in reverse. The function is divided into sections by differentiation, and the original function is then put back together via integration. The slope of a curve and the area under a curve are found geometrically using the differentiation and integration formulas, respectively.
Read More: Limits and Derivatives Formula
| Table of Content |
Key Terms: Differentiation, Integration, Calculus, Chain Rule, Antiderivatives, Definite Integral, Indefinite Integral, Integral calculus, Horizontal axis
What is Differentiation in Calculus?
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Differentiation can be defined as a derivative of a function related to the independent variable. It is used to measure the change in function in the independent variable per unit.
For example, if y = f(x) is a function of x, the rate of change of y per unit change in x is denoted by: dy/dx.
What is Integration in Calculus?
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Integration can be defined as the process to calculate definite or indefinite integrals. For a function and a closed interval on the real line, the definite integral is the area between the graph of the function, the horizontal axis, and the two vertical lines, which are at the endpoints of an interval. In the case of the absence of a specific integral, it is known as an indefinite integral.
The definite integral is calculated by using anti-derivatives. Thus, integration can be called the reverse process of differentiation. Differentiation is used to calculate the slope of a curve, whereas integration is used to calculate the area under the curve.
For example, for a function f(x) and a closed interval [a, b] on the real line, the definite integral is denoted by:
∫ba f(x) dx
Read More: Chain Rule Formula
Rules of Differentiation and Integration
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Some of the rules that are used for the differentiation of combinations of functions are the chain rule, product rule, and quotient rule. Different rules are utilized for the integration of functions such as the fundamental theorem of calculus, and the commonly used methods of integration namely, substitution method, integration by parts, integration by partial fractions, etc. The rules of differentiation and integration and their formulas are given below:
- Chain Rule of Differentiation: [f(g(x))]' = f'(g(x)) × g'(x)
- Product Rule of Differentiation: [f(x)g(x)]' = f'(x)g(x) + g'(x)f(x)
- Quotient Rule of Differentiation: [f(x)/g(x)]' = [f'(x)g(x) - g'(x)f(x)]/[g(x)]2
- The fundamental theorem of calculus 1: If f(x) is continuous over an interval [a,b] and the function F(x) is defined by
F(x) = \(\int_a^x{f (t)} {dt}\), then F'(x) = f(x)
- The fundamental theorem of calculus 2: The theorem states that if the anti-derivative of an integrated is given, then the definite integral can be evaluated at the endpoints of anti-derivative and subtracting.
\(\int_a^b{f(t)} {dt}\) = F(b) – F(a), where F(x) = \(\int_a^b{f (x)} {dx}\)
- Integration by Parts: ∫f(x)g(x) dx = f(x) ∫g(x) dx - ∫[f'(x) × ∫g(x) dx] dx
Integrals Detailed Video Explanation
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| Relevant Concepts | ||
|---|---|---|
| Methods of Integration | Integration by Partial Fractions | Integration by Parts |
| Cosine Rule | Roots of Polynomials | Degree of polynomial |
Differentiation and Integration Formulas
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The derivative of a function, f(x), is obtained by differentiating f(x), and the original function, f(x), is obtained by integrating f(x). In some cases, the reverse integration process is unable to produce the original function's constant terms, and as a result, the constant "C" is added to the integration's results.
List of Differentiation Formulas:
Differentiation formulas are tabulated below:
| SL No. | Differentiation Formulas |
|---|---|
| 1 | d/dx (c) = 0 |
| 2 | d/dx (x) = 1 |
| 3 | Constant Multiple Rule : d/dxc(x) = 1 |
| 4 | Sum or Difference Rule : d/dxf(x)+g(x)f(x)+g(x) = f’(x)+g’(x) d/dxf(x)−g(x)f(x)−g(x) = f’(x)-g’(x) |
| 5 | Power Rule: d/dx(xn) =nxn-1 , where n is any real number. |
| 6 | Natural Exponential Rule: d/dx(ex) = ex |
| 7 | Product Rule : d/dxf(x).g(x)f(x).g(x) = f’(x)g(x)+f(x)g’(x) |
| 8 | Quotient Rule : d/dxf(x)/g(x)f(x)/g(x) = f’(x)g(x)-f(x)g’(x) / g(x)g(x)2 |
| 9 | Chain Rule : ( fºg )( x ) equals f ′ ( g( x ) )·g′( x ) |
List of Integration Formulas:
Integration formulas are tabulated below:
| Integration Formulas | |
|---|---|
| ∫ xn dx | (xn+1/n+1) + C, where n ≠ -1 |
| ∫ sin x dx | - cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec2 x dx | tan x + C |
| ∫ cosec2 x dx | -cot x + C |
| ∫ sec x tan x dx | sec x + C |
| ∫ cosec x cot x dx | -cosec x +C |
| ∫ ex dx | ex + C |
| ∫ 1/x dx | ln x+ C |
| ∫11+x211+x2 dx | arctan x +C |
| ∫ ax dx | (ax/ln) a + C |
Read More: Differentiation Rules
Difference Between Differentiation and Integration
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The difference between differentiation and Integration are tabulated below:
| Differentiation | Integration |
|---|---|
| Differentiation is the process of calculating the rate at which one quantity changes in relation to another. | Integration is the process of combining smaller components into a single unit that functions as a single component. |
| It is used to determine the slope of a function at a point. | Integration is used to determine the integrated area under the curve of a function. |
| Derivatives are considered at a point. | Functions' definite integrals are taken into account over an interval. |
| Differentiation of a function is unique. | Integration may be not unique because of the arbitrary constant C. |
Read More: Dot Product Formula
Applications of Differentiation and Integration
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Integration can be defined in simple words as a mathematical approach of adding slices to find the whole. This theory of integration can be applied to easily find areas, volumes, median points, and other useful figures.
Applications of Differentiation are as follows:
- Differentiation helps calculate profits and losses using graphs
- It determines the temperature changes.
- It calculates the speed and distance traveled.
- Derivatives are assists in deriving equations.
- It helps find the range and magnitude of disasters during natural calamities.
Applications of Integration are as follows:
- Integration helps determine velocity, acceleration, and also distance.
- The average value of functions is computed using integration.
- Kinetic energy and work are determined with integration.
- Integration is utilized in calculating surface and probability.
Read More:
| Related Topics | ||
|---|---|---|
| Fundamental Theorem of Calculus | U Substitution Method | Reduction Formula in Integration |
| Differentiation and Integration Formula | Tan2x Formula | Definite Integral Formula |
Things to Remember
- Integration by parts is a unique mathematical technique of integration, mostly used in the case of two functions multiplied together.
- A derivative of a function connected to the independent variable is referred to as differentiation.
- Integration is the reverse process of differentiation.
- The integrated area under a function's curve can be found using integration.
- “C” is the arbitrary constant of integration.
- Differentiation enables the detection of the rate of change of velocity with respect to time (known as acceleration).
Previous Year Questions
- The value of sin2 51∘+ sin2 39∘ is… [KCET – 2020]
- If cos x = |sin x| then, the general solution is… [KCET – 2019]
- If 0≤ x< π/2, then the number of values of x… [JEE Main – 2019]
- A vertical lamp-post at the midpoint D… [JEE Main – 2019]
- If tanA+cotA=2, then the value of… [KCET – 2020]
- The value of cos245∘−sin215∘ is… [KCET – 2017]
- A, B and C are the angles opposite to the corresponding sides of lengths… [JKCET – 2017]
- The value of tan 8/π is equal to… [KCET – 2016]
- The value of tan10∘ tan20∘ tan30∘ tan40∘ tan50∘ tan60∘… [COMEDK UGET – 2012]
- A value of θ satisfying sin5θ−sin3θ+sinθ… [KCET – 2011]
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Sample Questions
Ques. Integrate the following with respect to x: ∫ x11 dx (2 Marks)
Ans. ∫ x11 dx = x (11 + 1)/(11 + 1) + c
= (x12/12) + c
Ques. Determine where, if anywhere, the function f(x) = x3+9x2−48x+2 is not changing. (3 Marks)
Ans. First the derivative,
f′(x)=3x2+18x−48=3(x2+6x−16)=3(x+8)(x−2)
Set this equation to zero and solve it.
f′(x) = 0 ⇒ 3(x+8)(x−2) = 0
The derivative will be zero at x= −8 and x=2. The function therefore not be changing at,
x= −8 and x=2
Ques. Find the tangent line to f(x) = 7x4+8x−6+2x at x=−1. (5 Marks)
Ans. The derivative of the function.
f′(x)=28x3−48x−7+2=28x3−48/x7+2
Next, evaluate the function and derivative at x=−1.
f(−1)=7+8−2=13
f′(−1)=−28+48+2=22
The equation of the tangent line.
y=f(−1)+f′(−1)(x+1)=13+22(x+1) → y=22x+35
Ques. Determine where the function h(z)=6+40z3−5z4−4z5 is increasing and decreasing. (5 Marks)
Ans. The derivative.
h′(z)=120z2−20z3−20z4=−20z2(z+3)(z−2)
Next, it is required to know where the function is not changing and set the derivative equal to zero and solve.
h′(z)=0 ⇒ −20z2(z+3)(z−2)=0
Hence, the derivative will be zero,
z=0 z=−3 z=2
To yield the result it is necessary to know where the derivative is positive (and hence the function is increasing) or negative (and hence the function is decreasing).
Because the derivative is continuous hence it can change sign where the derivative is zero. So, the number line determines the sign of the derivative for the various intervals.
Here is the number line for this problem. Therefore increasing/decreasing information.
Increasing: −3
Decreasing: −∞
Ques. Determine where, if anywhere, the tangent line to f(x)=x3−5x2+x is parallel to the line y=4x+23. (5 Marks)
Ans.The derivative of the function.
f′(x)=3x2−10x+1
Two lines that are parallel will have the same slope and so the slope of the tangent line will be 4, the slope of the given line.
f′(x)=4 → 3x2−10x+1=4 → 3x2−10x−3=0
x = (10±√136)/6 = (10±2√34)/6 = (5±√34)/3
So, the tangent line will be parallel to y=4x+23y=4x+23 at,
x = (5−√34)/3 = −0.276984
x = (5+√34)3 = 3.61032
Ques. Find \(\int\) \( sin^2 x-cos^2x \over sin^2xcos^2x\)(5 Marks)
Ans. Let, l = \(\int\)\(\frac{dx}{sin^2cos^2x}\)
On dividing the numerator and denominator by cos4 x,
l = \(\int\)\(\frac{sec^2 x. sec^2 x}{tan^2x}dx\)
On dividing the numerator and denominator by cos4 x,
l = \(\int \frac{sec^2x.sec^ex}{tan^2x}dx\)
⇒ l = \(\int \frac{(1 + tan^2x).sec^ex}{tan^2x}dx\)
Put tan x = t ⇒ sec2xdx = dt
∴ l = \(\int \frac{1+t^2}{t^2}dt = \int 1dt + \int \frac{1}{t^2}dt\)
⇒ l = t – \(\frac{1}{t}\) + C
⇒ l = tan x – cot x +C
Ques. Evaluate \(\int cos^-1(sinx)dx\) (2 Marks)
Ans. Let I = ∫ cos-1 (sin x) dx
= ∫ cos-1 [ cos (π/2 - x) ] dx
= ∫ (π/2 - x) dx
= π/2 ∫ dx - ∫ xdx = π/2 x - x2/2 + C
Ques. Write the anti-derivative of: (3√x + 1/√x). (3 Marks)
Ans. The anti-derivative of (3√x + 1/√x)
Anti-derivative of \((3 \sqrt{x} + \frac{1}{\sqrt{x}})\)
= ∫ \((3 \sqrt{x} + \frac{1}{\sqrt{x}})\) dx = ∫3 \(\sqrt{x}\)dx + ∫\(\frac{1}{\sqrt{x}}\) dx
= 3 (\(\frac{x^{1/2 +1}}{1/2 + 1}\)) = [\(\frac{x^{-1/2 +1}}{-1/2 + 1}\)] + C
= 2 (x3/2 + x1/2) + C
Ques. Evaluate ∫ sec2 (7 - 4x) dx. (2 Marks)
Ans.
Let l = ∫ sec2 (7 – 4x)dx
= \(\frac{tan(7-4x)}{-4} +C\) [\(\because\) sec2 axdx = \(\frac{tan ax}{a}\)]
= – \(\frac{tan(7-4x)}{-4} +C\)
Ques. Mention any two applications of differentiation. (2 Marks)
Ans. Applications of Differentiation are as follows:
- It calculates the speed and distance traveled.
- Derivatives are assists in deriving equations.
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