Indefinite Integral Calculator

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.

• Take any function to compute the indefinite integral.
• Go through the different rules like power rule, exponential, constant rule, etc before solving the problem.
• ∫ x dx is always equal to (x²) / 2 + C. Where C is the constant
• Integration of any constant is equal to the constant value * x + C.
• If the function is in the difficult form.
• Consider one part of the function as a variable and substitute that variable in all the possible places of the fumction.
• Find integration with respect to that variable and substitute the value.

Examples

Question 1: Solve ∫ (2x + 1 ) / (x+5)³ dx?

Solution:

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/ a^b = a^-b

= 2 ∫ u^-2 du – 9 ∫ u^-3 du

Apply the Power rule: ∫ x^a dx = x^a+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?

Solution:

∫ (x² + 3x – 2) dx

= ∫ x² dx + ∫ 3x dx – ∫ 2 dx

= x³/3 + (3x)² / 2 – 2x + C

= x³/3 + 9x²/2 – 2x + C

Find a variety of Other free Maths Calculators that will save your time while doing complex calculations and get step-by-step solutions to all your problems in a matter of seconds.

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.

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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.

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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.

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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.

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