Anti-derivative Calculator

Anti-derivative Calculator finds the integral value of a function. The process of finding the anti-derivative of a function is known as integration. In other words, the reverse process of differentiation is called integration. Anti-derivative is also known as the integral of a function.

What is Anti-derivative Calculator?

Anti-derivative Calculator is an online tool used to calculate the value of a given indefinite integral. Integration can be used to find the area under a curve. It can also be used to determine the volume of a three-dimensional solid shape. To use the anti-derivative calculator, enter the function in the input box.

Anti-derivative Calculator

How to Use Anti-derivative Calculator?

Please follow the simple steps to find the anti-derivative of a function using the anti-derivative calculator:

• Step 1: Go to Cuemath's online anti-derivative calculator.
• Step 2: Enter the function in the input box of the anti-derivative calculator.
• Step 3: Click on the "Calculate" button to find the anti-derivative of the function.
• Step 4: Click on the "Reset" button to clear the fields and enter new values.

How Does Anti-derivative Calculator work?

There are many uses of integration. The average value of a curve, the area between two curves, the center of gravity, and the center of mass can all be determined by using integration. There are two types of integrals available in calculus. These are as follows:

• Indefinite Integrals - Such integrals do not have specified limits thus, the final value of the integral is indefinite. If we integrate the derivative of a function say g'(x), we will get the function itself.

• Definite Integrals - Integrals that have defined limits with pre-existing values are known as definite integrals. Such an integral is used to find the area under a curve between two given points (these points act as the limits).

Given below are some properties of integration.

• Differentiating the integral will result in the integrand; ∫ f(x) dx = f(x) + C.
• If two indefinite integrals have the same derivative then they will be equivalent. This is because the two integrals lead to the same family of curves; ∫ [ f(x) dx - g(x) dx] =0
• When we perform integration, the coefficient of the variable is taken outside the integral sign; ∫ k f(x) dx = k ∫ f(x) dx.

The formula for determining the value of a simple integral is given as: ∫xⁿ dx = (xⁿ⁺¹ / n+1) + C.

Solved Examples on Anti-derivatives

Example 1: Find the anti-derivative value of 5x³ + 2x² and verify it using the anti-derivative calculator.

Solution:

Using the formula: ∫xⁿ dx = (xⁿ⁺¹ / n+1) + C

= ∫( 5x³ + 2x²) dx

= ∫( 5x³) dx + ∫(2x²) dx

= [5 × (x^{3 + 1} / 3 + 1)] + [2 × x^{2 + 1} / 2 + 1]

= 5x⁴ / 4 + 2x³ / 3.

Therefore, the anti-derivative value of 5x³ + 2x² is 5x⁴ / 4 + 2x³ / 3.

Example 2: Find the anti-derivative value of 1 + x⁴ and verify it using the anti-derivative calculator.

Solution:

Using the formula: ∫xⁿ dx = (xⁿ⁺¹ / n+1) + C

= ∫( 1 + x⁴) dx

= ∫( 1.x⁰) dx + ∫(x⁴) dx

= [1 × (x^{0 + 1} / 0 + 1)] + [x^{4 + 1} / 4 + 1]

= x + x⁵/ 5

Thus, the anti-derivative of 1 + x⁴ is x + x⁵/ 5

Similarly, you can use the anti-derivative calculator to find the value of anti-derivatives for the following:

• x³ / 2
• 5x² + 6x

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