Solve Indefinite integral :The indefinite integral of a function  f(x) is a family of functions  F(x) such that the derivative of  F(x) equals  f(x). In other words:

The indefinite integral is written as:

Where

• F(x) is the integrand ( the function being integrated)
• dx indicates the variable of integration.
• F(x) is the antiderivative of f(x).
• The family of all potential antiderivatives is represented by the constant of integration, C.

Solve Indefinite integral math problems

Example 1

Step 1: Expand sin3x using the angle formula

We are aware of the expansion for sin(3x) ⁡.

Substitute this in the integral

Expand the integral

Step 2: Solve the first term

For 3∫cos^4(x)sin(x)dx, use substitution: Let u=cos(x), so du=−sin⁡(x)dx This becomes:

Substitute back u=cosx

Step 3: Solve the second term

For −4∫cos^⁡4(x)sin⁡3(x) dx, use

So the integral becomes

Substitute u=cos⁡(x), du=−sin⁡(x)dx:

Simplify

Integrate

Substitute back u=cos(x)

Step 4: Combine results

Add the two terms together

Simplify

Final answer

Example 2

Step 1: Use the half-angle identity

We know the identity

Substitute this into the integral

Simplify the square root

Factor out √2

Step 2: Substitution

Let u=x/2, so x=2u and dx=2 du. Substitute:

Simplify

Step 3: Integrate

The integral of cos (u) is sin(u)

Step 4: Back substitute

Recall that u=x/2​, so:

Final answer

Example 3

Step 1: Use the product to sum identities

The product of two cosines can be expressed as

Apply this identity to cos2xcos3x:

Since cos⁡(−𝑥) = cos⁡𝑥 this can be simplified to:

Step 2: Substitute into the original integral

Now substitute this result into $\int \cos x\cos 2x\cos 3xdx$:

Simplify

Step 3: Solve term separately.

a) First term: ∫cos⁡xcos⁡5x dx

Use the product-to-sum identity again:

This becomes: as cos(−4x)=cos4x

Substitute into the integral:

Evaluate

Thus

b) Second term: ∫cos²x dx

Use the identity cos²x= (1+cos2x)/2

Evaluate

Thus

Step 4: Combine the results.

Combine the outcomes for the two terms now:

Simplify

Final answer

With practice, you’ll discover that solving integrals is a natural and gratifying process! Here are some basic integral rules

Some practice problems

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What are domain and range?

Determine the domain and range

Domain

A function’s domain refers to the set of all potential input values (x-values) for which the function is defined.

General Steps:
1. Start with all real numbers:

Assume  x can be any real number (unless restricted).

2. Identify the restrictions:

Check for cases where the function is undefined:

• Divide by zero. If the denominator is zero, remove those  x-values.
• Square Roots: For even roots, ensure the radicand (expression inside the root) is greater than or equal to zero.
• Logarithms: require an argument greater than or equal to zero.

Express the domain:

• Use interval notation or inequalities to describe the domain.

Range:

The range of a function is the set of all potential output values (y-values).

General Steps:

Analyze function behavior:

Determine how 𝑦  (the output) changes as x varies over the domain.
Consider the restrictions:

If any y-values are impossible to achieve, remove them from the range.
Solve for y:

Consider expressing  x in terms of y (inverse function) and identifying any limits on  y.

How to find domain and range of a function?

Example 1:

Determine the domain and range of the following function

Domain:

The domain is the set of all  x-values for which the function is defined.

Restrictions:

1.Square root restriction: The expression inside the square root, 9-x^2, must be ≥0

2. Denominator restriction: The denominator cannot equal 0. Regarding the square root in the denominator:

Domain: Combine these restrictions: -3 < 𝑥 < 3 .
In interval notation: 𝑥 ∈ ( – 3, 3) .

Range:

The range of a function refers to its possible y-values.

Analysis:

Range of y:

y values range from 1 to ∞ (excluding infinity).
Range: In interval notation:  y∈( 1/3 ​,∞).

Final Answer:

• Domain: $x\in (-3,3)$
• Range: y∈(1/3,∞)

Example 2:

Determine the domain and range of the following function

1. Domain: The domain is the set of all x-values for which the function is defined.

Restrictions:

Square root restriction: The expression inside the square root,  x ^2 −7x+10, must be non-negative (i.e., ≥ 0 ≥0), because the square root of a negative number is undefined in the real number system.

To solve this inequality, first factor the quadratic expression:

The inequality becomes:

To solve this, identify the critical locations when the expression equals zero.

We evaluate the sign of  (x−2)(x−5) in the intervals specified by the crucial points  x=2 and x=5.

For x < 2, both (𝑥 – 2) and (𝑥 – 5) are negative, resulting in a positive product.

For 2 < 𝑥 < 5 , ( 𝑥 − 2 ) is positive and (x−5) is negative, resulting in a negative product.

For 𝑥 > 5, both (𝑥 – 2) and (𝑥 – 5) are positive, indicating that the product is positive.
The inequality ( 𝑥 − 2 ) ( 𝑥 − 5 ) ≥ 0

Domain:

• The domain is x∈(−∞,2]∪[5,∞)

2. Range

The range is the set of all possible y-values.

Analysis:

Since the square root function always produces non-negative outputs, 𝑦 ≥ 0

The smallest value of  y happens when  x=2 or x=5, as  x ^2 −7x+10=0. At these places, 𝑦 = 0.

As $x$ moves away from x=2 or $x=5$ (in either direction), $y$ increases because the expression inside the square root becomes positive.

Range:

Since  y≥0 and y can grow unbounded as  x travels away from 2 and 5, the range is:

Final Answer:

Domain: 𝑥 ∈ ( −∞, 2 ]∪ [ 5 , ∞ )
Range: 𝑦 ∈ [ 0 , ∞ )

Some related practice problems

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Derivative using first principle

How do I find the derivative using the first principle of sinx ?

To find the derivative of $\sin (x)$ using the first principles (also known as the definition of the derivative), we start with the general formula:

​

Here, f(x)=sin(x). So, applying the first principle:

Step 1: Calculate sin(x+Δx) using the Sum of Angles Formula.
For sine, the sum of angles identity is:

Substitute this into the expression:

Step 2: Make the expression simpler
Combine the phrases now:

Step 3: Divide the upper limit
The limit can be divided into two sections:

Step 4: Evaluate each limit

The first limit

The second limit,

Step 5: Combine the outcomes
After the application of the limits:

This simplifies to:

Final Answer:

The derivative of $\sin (x)$ using the first principles is:

How do I find the derivative of sinx by first principle?

Derivative using first principle

Here, f(x)=cos⁡−1(x) so we need to compute the derivative as

Step 1: Apply the identity for inverse trigonometric functions.

The identity can be used to calculate the difference between two inverse cosine functions:

However, this identity may not immediately simplify the phrase, so let us proceed with the chain rule technique.

Step 2: Chain Rule for Derivatives

To calculate the derivative of  cos −1 (x), we can use the following formula:

We may confirm this finding using first principles, as follows:

Step 3: Create the expression from first principles.

Let f(x)=cos−1(x), and we want to compute:

We use the approximation that for small h

where the derivative of $)$ is known.

In conclusion, we confirm the derivative of cos ⁡ −1 (x) by applying the limit:

So, the derivative of cos−1(x) is: