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

1.

2.

3.

4.

5.

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:

Successive differentiation is an important concept in calculus that is used to study the behavior of functions, such as their rates of change, curvature, and other dynamic characteristics.

What is successive differentiation?

Successive differentiation is the process of differentiating a function several times to produce higher-order derivatives. The first derivative,  f ′ (x), describes the rate of change of the original function f(x). Repeated differentiation provides the second derivative f ′′ (x), the third derivative f ′′′ (x), and so on, providing a better knowledge of the function’s behavior.

Notation for successive differentiation:

1. First Derivative: The first derivative of a function y=f(x) with respect to $x$ is denoted as:

2. Second Derivative: The second derivative, or derivative of the first derivative, is expressed as:

3. Third Derivative: The third derivative, the derivative of the second derivative, is represented by:

4. Higher-order Derivatives: Derivatives of order  n, where  n is a positive integer bigger than 3, are notated as:

Here, f^{(n)}(x) uses a superscript in parentheses to denote the $n$-th derivative.

Example 1:

• (first derivative)
• (second derivative)
• (third derivative)
• (fourth derivative):

Problem 1: If $y=\sin (ax+b)$, then show that

Step 1: First Derivative

Let y=sin(ax+b).The first derivative is:

Step 2: Second derivative

Step 3: Third derivative

Step 4: Fourth derivative

Step 5: Pattern Identification

From the derivatives, we notice a pattern

First derivative:

Second derivative:

Third derivative:

Fourth derivative:

The n-th derivative is:

Successive differentiation is a strong calculus tool with numerous applications in science, engineering, and other fields. Understanding its notation and concepts enables us to study complicated systems, optimize processes, and acquire profound insights into the dynamics of functions.

Understanding successive differentiation provides you with a solid mathematical foundation, opening the door to further investigation and practical applications in a variety of domains. So keep discriminating and finding!