Publisher : Nina Watson
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 10x4 + 6sin(x) – 12x5y3 + 2xy w.r.t “x”.
Solution:
Step 1: Apply the integral to the function.
ʃ (10x4 + 6sin(x) – 12x5y3 + 2xy) dx
Step 2: Separate the integrals using the sum rule.
ʃ (10x4 + 6sin(x) – 12x5y3 + 2xy) dx = ʃ (10x4) dx + ʃ 6sin(x) dx – ʃ 12x5y3 dx + ʃ 2xy dx
Step 3: Apply multiplication by constant rule and write the constants outside of the integrals.
ʃ (10x4 + 6sin(x) – 12x5y3 + 2xy) dx = 10 ʃ (x4) dx + 6 ʃ sin(x) dx – 12 y3 ʃ x5 dx + 2y ʃ x dx
Step 4: Simplify.
ʃ (10x4 + 6sin(x) – 12x5y3 + 2xy) dx = 10 (x4+1 / 4 + 1) + 6 (– cos(x)) – 12y3(x5+1 / 5 + 1) + 2y (x1+1 / 1 + 1) + C
= 10 (x5 / 5) – 6 (cos(x)) – 12y3 (x6 / 6) + 2y (x2 / 2) + C
= 10/5 (x5) – 6 (cos(x)) – (x6) 12y3 / 6 + 2y (x2 / 2) + C
= 2 (x5) – 6 (cos(x)) – 2y3 (x6) + y (x2) + C
= 2x5 – 6 cos(x) – 2 x6y3 + x2y + C
To get rid of these long calculations use an integral calculator. This calculator will give you a step-by-step solution to the integral problems in a couple of seconds.Example 2: For definite integrals
Integrate 22x5 + 3sin(x) – 6x3 + 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 : In 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.