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Integration Made Easy

By Masood Amir

Mathematics Worksheets, ASVA provide calculus and Algebra worksheets.

Calculus:

To measure change or variation of a function with respect to the independent variable we use Calculus.

Differential Calculus: (Mathematics Worksheets)

Differrential Calculus used to measure change or variation of a function in a very small invterval of time.

Integral Calculus:

The branch of calculus used to measure changes or variation over an interval of independent variable, called Integral calculus, e.g to find length of curve, the area of region and the volume of a solid in a specified period of time.

The technique used to measure such changes or variation called “Integration” or “Antiderivatives”. It a reverse process of differentiation.

Mathematically, Integration defined as “ If $f'(x)$ represents the differential coefficient of $f(x)$ , we need to find $f(x)$ , if we have $f'(x)$ or $dy/dx$ .

Integration

Notation: $''∫''$ used to show the integration, it is a symbol of $``S''$ derived from the word $``Sum''$ . i.e. Integration is a process in which we have to sum up the derivatives over a specified interval and to find the function.

Techniques of Integration:

As we know that integration is the reverse process of differentiation, our problem is to find the function $f(x)$ or $Y$ , when $f'(X)$ or $dy/dx$ is given.

$dy/dx &= f'(X)$

$∫dy &= ∫f '(X)dx$

$Y &= f(x)$ is our solution

Ist Formula of Integration (1st Rule of Integration)

Indefinite Integration:

Ist Formula of Integration (Ist Rule of Integration):

Let

$\int dy = \int x^n dx$

$y = \frac{x^{n+1}}{n+1} + C$

Why $``C''$ :

In the process of differentiation, we eliminate constant, as the derivative of a constant is “ $zero$ “.

So, In functions like $x^n$ , $x^n + 6$ , $x^n - 3$ , $x^n + k$ the derivatives of all of them is $X^{n-1}$ , in finding the anti derivative of $X^{n-1}$ , we put a constant “ $C$ ”, as we don’t know which constant was present in the original function, and can be found If we have initial boundary values (Definite Integral).

Example: Solve $\int x^n \, dx$

Solution:

$\int dy = \int x^3 dx$

$y = \frac{x^{3+1}}{3+1} + C$

$y = \frac{x^{4}}{4} + C$

Example: Solve

$\int (x^3 + x^2 + 5x + 6) \, dx$

Solution:

$\int (x^3 + x^2 + 5x + 6) \, dx$

$\int x^3 \, dx + \int x^2 \, dx + \int 5x \, dx + \int 6 \, dx$

$\frac{x^4}{4} + \frac{x^3}{3} + \frac{5x^2}{2} + 6x + C$

Worksheet # 1

Find the Integral of the following:

SET 1

(1) $\begin{align*} 1. & \quad \int (x^3 - 4x^2 + 5x - 6) \, dx \\ 2. & \quad \int (3x^5 - 4x^3 + 3x^2) \, dx \\ 3. & \quad \int (ax^5 - bx^4) \, dx \\ 4. & \quad \int \left( \frac{x^3}{2} - \frac{5x^4}{3} + 3x^2 \right) \, dx \\ 5. & \quad \int (4\sqrt{3}x^2 - 2x) \, dx \\ 6. & \quad \int \left( 3x + 5x^2 - \frac{x^3}{2} - 0.4x^4 \right) \, dx \\ 7. & \quad \int \left( x(8x - \frac{1}{2}) \right) \, dx \\ 8. & \quad \int (2 - x)(4 + 3x) \, dx \\ 9. & \quad \int \left( x^{-3} + x^{-4} \right) \, dx \\ 10. & \quad \int \left( (2x^3 - 3)(3x^4) \right) \, dx \\ 11. & \quad \int \left( 4x^7 + 3x^{12} - 5x^8 + 2x - 1 \right) \, dx \\ 12. & \quad \int \left( ax^3 - bx^2 + cx - d \right) \, dx \\ 13. & \quad \int \left( \frac{1}{x^3} + \frac{2}{x^2} - 6 \right) \, dx \\ 14. & \quad \int \left( -3x - 8 + 2\sqrt{x} \right) \, dx \\ 15. & \quad \int \left( (x^3 - 5)(2x + 5) \right) \, dx \\ 16. & \quad \int \left( 7x - 6 + 5\sqrt{x} \right) \, dx \end{align*}$

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