Introduction to integral : Definition, Types, and examples.

The process of integration is the opposite of derivation/differentiation. We use the symbol “∫” to apply integration to any function. This notation represents the sum of many small quantities. In this article, we will learn the definition and types of integrals with the help of examples.

The process of integration is the opposite of derivation/differentiation. We use the symbol “∫” to apply integration to any function. This notation represents the sum of many small quantities. In this article, we will learn the definition and types of integrals with the help of examples.

Examples are the most important part of any topic it helps us to understand the concepts more clearly.

**Example 1 : For indefinite integral**

Evaluate the integration of 10x^{4} + 6sin(x) – 12x^{5}y^{3} + 2xy w.r.t “x”.

**Solution: **

**Step 1:** Apply the integral to the function.

ʃ (10x^{4} + 6sin(x) – 12x^{5}y^{3} + 2xy) dx

**Step 2:** Separate the integrals using the sum rule.

ʃ (10x^{4} + 6sin(x) – 12x^{5}y^{3} + 2xy) dx = ʃ (10x^{4}) dx + ʃ 6sin(x) dx – ʃ 12x^{5}y^{3} dx + ʃ 2xy dx

**Step 3:** Apply multiplication by constant rule and write the constants outside of the integrals.

ʃ (10x^{4} + 6sin(x) – 12x^{5}y^{3} + 2xy) dx = 10 ʃ (x^{4}) dx + 6 ʃ sin(x) dx – 12 y^{3 }ʃ x^{5} dx + 2y ʃ x dx

**Step 4:** Simplify.

ʃ (10x^{4} + 6sin(x) – 12x^{5}y^{3} + 2xy) dx = 10 (x^{4+1} / 4 + 1) + 6 (– cos(x)) – 12y^{3}(x^{5+1} / 5 + 1) + 2y (x^{1+1} / 1 + 1) + C

= 10 (x^{5} / 5) – 6 (cos(x)) – 12y^{3 }(x^{6} / 6) + 2y (x^{2} / 2) + C

= 10/5 (x^{5}) – 6 (cos(x)) – (x^{6}) 12y^{3 }/ 6 + 2y (x^{2} / 2) + C

= 2 (x^{5}) – 6 (cos(x)) – 2y^{3 }(x^{6}) + y (x^{2}) + C

= 2x^{5} – 6 cos(x) – 2 x^{6}y^{3} + x^{2}y + C

**Example 2: For definite integrals **

Integrate 22x^{5} + 3sin(x) – 6x^{3} + 14x w.r.t **x, **on the interval **[2, 5]**.

**Solution **

**Step 1:** Apply the integral symbol on the function, and write the limits carefully.

**Step 2:** Apply the sum rule.

**Step 3:** Write the constants outside the integrals using the constant multiplication rule.

**Step 4:** Simplify.

**Step 5:** Put the limits in the variable x.

**Summary : I**n this article, we studied the definition of integral and types of integrals along with examples. Now you are witnessed that integration is not a difficult topic. After reading this article you can solve the pre-calculus integrals easily.

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