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(sinx+cosx)²

Integral of (sinx+cosx)² dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1                      
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 |                   2   
 |  (sin(x) + cos(x))  dx
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0                        
$$\int\limits_{0}^{1} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}\, dx$$
Integral((sin(x) + cos(x))^2, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of a constant is the constant times the variable of integration:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Let .

            Then let and substitute :

            1. The integral of a constant times a function is the constant times the integral of the function:

              1. The integral of cosine is sine:

              So, the result is:

            Now substitute back in:

          So, the result is:

        The result is:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. There are multiple ways to do this integral.

          Method #1

          1. Let .

            Then let and substitute :

            1. The integral of a constant times a function is the constant times the integral of the function:

              1. The integral of is when :

              So, the result is:

            Now substitute back in:

          Method #2

          1. Let .

            Then let and substitute :

            1. The integral of is when :

            Now substitute back in:

        So, the result is:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Let .

            Then let and substitute :

            1. The integral of a constant times a function is the constant times the integral of the function:

              1. The integral of cosine is sine:

              So, the result is:

            Now substitute back in:

          So, the result is:

        1. The integral of a constant is the constant times the variable of integration:

        The result is:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of a constant is the constant times the variable of integration:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Let .

            Then let and substitute :

            1. The integral of a constant times a function is the constant times the integral of the function:

              1. The integral of cosine is sine:

              So, the result is:

            Now substitute back in:

          So, the result is:

        The result is:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Let .

            Then let and substitute :

            1. The integral of a constant times a function is the constant times the integral of the function:

              1. The integral of cosine is sine:

              So, the result is:

            Now substitute back in:

          So, the result is:

        1. The integral of a constant is the constant times the variable of integration:

        The result is:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       
 |                                        
 |                  2                 2   
 | (sin(x) + cos(x))  dx = C + x - cos (x)
 |                                        
/                                         
$$\int \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}\, dx = C + x - \cos^{2}{\left(x \right)}$$
The graph
The answer [src]
   2           2   
cos (1) + 2*sin (1)
$$\cos^{2}{\left(1 \right)} + 2 \sin^{2}{\left(1 \right)}$$
=
=
   2           2   
cos (1) + 2*sin (1)
$$\cos^{2}{\left(1 \right)} + 2 \sin^{2}{\left(1 \right)}$$
cos(1)^2 + 2*sin(1)^2
Numerical answer [src]
1.70807341827357
1.70807341827357
The graph
Integral of (sinx+cosx)² dx

    Use the examples entering the upper and lower limits of integration.