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Indefinite Integral Calculator: Do you feel calculating indefinite integral somewhat difficult? Not anymore with the help of our easy to use online calculator tool. Now, you can solve the integration of any function easily and instantly by using our Indefinite Integral Calculator. Refer the below section, to get familiar with this concept by checking the solved examples. The step by step process to compute the indefinite integral is also mentioned here.
Indefinite integral is an integral without having upper and lower limits. The step by step process of evaluating indefinite integrals is listed here. So, follow the steps given here and do calculations easily by hand.
Question 1: Solve ∫ (2x + 1 ) / (x+5)3 dx?
Given input Mixed Number is 2 4/3
∫ (2x + 1 ) / (x+5)3 dx
Let us take,
u = x + 5
Then, 2x + 1 = 2u - 9
∫ (2x + 1 ) / (x+5)3 dx = ∫ (2u - 9) / u3 du
= ∫ 2u / u3 - 9/ u3 du
= ∫ 2/u2 - 9/u3 du
Apply the sum rule ∫ f(x) + g(x) dx = ∫ f(x) dx + ∫ g(x) dx
= ∫ 2/u2 du - ∫ 9/u3 du
Take out the constant: ∫ a. f(x) dx = a. ∫ f(x) dx
= 2 ∫ 1/ u2 du - 9 ∫ 1 / u3 du
Apply exponent rule 1/ ab = a-b
= 2 ∫ u-2 du - 9 ∫ u-3 du
Apply the Power rule: ∫ xa dx = xa+1 / a+1, a ≠ 1
= 2 * u-2+1 / (-2+1) - 9 * u-3+1 / (-3+1)
= 2 * u-1 / (-1) - 9 * u-2 / (-2)
= -2/u + 9/2u2
Substitute u = x + 5 in the above
= -2/(x + 5) + 9/2( x + 5)2
∫ (2x + 1 ) / (x+5)3 dx = -2/(x + 5) + 9/2( x + 5)2 + C
Question 2: Solve ∫ (x2 + 3x – 2) dx?
∫ (x2 + 3x – 2) dx
= ∫ x2 dx + ∫ 3x dx – ∫ 2 dx
= x3/3 + (3x)2 / 2 - 2x + C
= x3/3 + 9x2/2 - 2x + C
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1. What is Meant by Indefinite Integrals?
Indefinite Integral is an integration function indicated without lower and upper limits and with an arbitrary constant C. It is considered as an easy way to symbolize the antiderivative of the function. The representation is ∫ f (x) dx. The function f(x) is called Integrand.
2. Why we add a constant with an Indefinite Integral?
Integral function is called its anti derivative. If you differentiate a function and then integrate it, you should get the function back.
For example take f(x) = x, g(x) = x + 4
f′(x)=1 and g'(x) = 1.
If you not add constant
∫f′(x) dx = x, ∫g'(x) dx = x
Here, you are not getting exact g(x) value. So, we are adding constant.
∫f′(x) dx = x + C1, ∫g'(x) dx = x + C2
C1 = 0, C2 = 4.
3. What are the rules of Integration?
Integration is used to find area, volume, etc. The some of the common rules of integration are:
1. Constant rule: ∫a dx = ax + C
2. Multiplication by constant: ∫cf(x) dx = c ∫f(x) dx
3. Reciprocal rule: ∫(1/x) dx = log(x) + C
4. Exponential rules: ∫ex dx = ex + C, ∫ax dx = ax/log(a) + C, ∫log(x) dx = x log(x) − x + C.
4. Can an Integral have 2 Answers?
No, integrals don't have two answers. Let us say, x+c, x2+c both cannot be solutions to same integral, because x and x2 don't differ by constant.