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Integrals - Some properties of indefinite integral

Grade 12CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The process of differentiation and integration are inverses of each other. This is expressed as ddx∫f(x)dx=f(x)\frac{d}{dx} \int f(x) dx = f(x) and ∫f′(x)dx=f(x)+C\int f'(x) dx = f(x) + C, where CC is the constant of integration.

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Two indefinite integrals with the same derivative lead to the same family of curves and are considered equivalent. This implies that if ddx∫f(x)dx=ddx∫g(x)dx\frac{d}{dx} \int f(x) dx = \frac{d}{dx} \int g(x) dx, then ∫f(x)dx=∫g(x)dx+C\int f(x) dx = \int g(x) dx + C.

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The integral of the sum or difference of two functions is the sum or difference of their respective integrals: ∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx.

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A constant factor can be moved outside the integral sign: ∫k⋅f(x)dx=k∫f(x)dx\int k \cdot f(x) dx = k \int f(x) dx for any real number kk.

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The linearity property allows us to integrate a linear combination of functions: ∫[k1f1(x)+k2f2(x)+...+knfn(x)]dx=k1∫f1(x)dx+k2∫f2(x)dx+...+kn∫fn(x)dx\int [k_1 f_1(x) + k_2 f_2(x) + ... + k_n f_n(x)] dx = k_1 \int f_1(x) dx + k_2 \int f_2(x) dx + ... + k_n \int f_n(x) dx.

📐Formulae

ddx∫f(x)dx=f(x)\frac{d}{dx} \int f(x) dx = f(x) Pach

∫f′(x)dx=f(x)+C\int f'(x) dx = f(x) + C

∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx

∫kf(x)dx=k∫f(x)dx\int k f(x) dx = k \int f(x) dx

∫[k1f1(x)+k2f2(x)]dx=k1∫f1(x)dx+k2∫f2(x)dx\int [k_1 f_1(x) + k_2 f_2(x)] dx = k_1 \int f_1(x) dx + k_2 \int f_2(x) dx

💡Examples

Problem 1:

Evaluate the integral: ∫(4x3+3x2+2x+5)dx\int (4x^3 + 3x^2 + 2x + 5) dx

Solution:

∫(4x3+3x2+2x+5)dx=∫4x3dx+∫3x2dx+∫2xdx+∫5dx\int (4x^3 + 3x^2 + 2x + 5) dx = \int 4x^3 dx + \int 3x^2 dx + \int 2x dx + \int 5 dx =4∫x3dx+3∫x2dx+2∫xdx+5∫1dx= 4 \int x^3 dx + 3 \int x^2 dx + 2 \int x dx + 5 \int 1 dx =4(x44)+3(x33)+2(x22)+5x+C= 4 \left( \frac{x^4}{4} \right) + 3 \left( \frac{x^3}{3} \right) + 2 \left( \frac{x^2}{2} \right) + 5x + C =x4+x3+x2+5x+C= x^4 + x^3 + x^2 + 5x + C

Explanation:

We apply the property of the integral of a sum and the property of scalar multiplication. Each term is integrated individually using the power rule ∫xndx=xn+1n+1\int x^n dx = \frac{x^{n+1}}{n+1}.

Problem 2:

Find the integral of: ∫(2sin⁡x−3cos⁡x+ex)dx\int (2 \sin x - 3 \cos x + e^x) dx

Solution:

∫(2sin⁡x−3cos⁡x+ex)dx=∫2sin⁡xdx−∫3cos⁡xdx+∫exdx\int (2 \sin x - 3 \cos x + e^x) dx = \int 2 \sin x dx - \int 3 \cos x dx + \int e^x dx =2∫sin⁡xdx−3∫cos⁡xdx+∫exdx= 2 \int \sin x dx - 3 \int \cos x dx + \int e^x dx =2(−cos⁡x)−3(sin⁡x)+ex+C= 2(-\cos x) - 3(\sin x) + e^x + C =−2cos⁡x−3sin⁡x+ex+C= -2 \cos x - 3 \sin x + e^x + C

Explanation:

Using the linearity property, we split the integral into three parts and move constants outside. We then apply standard trigonometric and exponential integration rules.

Problem 3:

Given ddxF(x)=1x\frac{d}{dx} F(x) = \frac{1}{x}, find ∫1xdx\int \frac{1}{x} dx.

Solution:

By the property ∫f′(x)dx=f(x)+C\text{By the property } \int f'(x) dx = f(x) + C Since ddx(log⁡∣x∣)=1x\text{Since } \frac{d}{dx}(\log|x|) = \frac{1}{x} ∫1xdx=log⁡∣x∣+C\int \frac{1}{x} dx = \log|x| + C

Explanation:

This demonstrates the inverse relationship between differentiation and integration. Since the derivative of log⁡∣x∣\log|x| is 1x\frac{1}{x}, the indefinite integral of 1x\frac{1}{x} is log⁡∣x∣+C\log|x| + C.