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Integral of (8x⁵-3÷sin²4x+2÷3x²)dx dx

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The solution

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 |  /                     2\   
 |  |   5      3       2*x |   
 |  |8*x  - -------- + ----| dx
 |  |          24       3  |   
 |  \       sin  (x)       /   
 |                             
/                              
0                              
$$\int\limits_{0}^{1} \left(\frac{2 x^{2}}{3} + \left(8 x^{5} - \frac{3}{\sin^{24}{\left(x \right)}}\right)\right)\, dx$$
Integral(8*x^5 - 3/sin(x)^24 + 2*x^2/3, (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    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:

    1. Integrate term-by-term:

      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:

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

        1. Rewrite the integrand:

        2. There are multiple ways to do this integral.

          Method #1

          1. Rewrite the integrand:

          2. Integrate term-by-term:

            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:

            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. 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. 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. 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. 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. 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. 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. 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. 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. 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:

            The result is:

          Method #2

          1. Rewrite the integrand:

          2. Integrate term-by-term:

            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:

            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. 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. 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. 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. 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. 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. 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. 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. 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. 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:

            The result is:

        So, the result is:

      The result is:

    The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                                                                                                                                                    
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 | /                     2\                                                                                             3        23         6         21             19             7             17              13   
 | |   5      3       2*x |                           3            5            15             9             11      2*x    3*cot  (x)   4*x    11*cot  (x)   165*cot  (x)   495*cot (x)   495*cot  (x)   1386*cot  (x)
 | |8*x  - -------- + ----| dx = C + 3*cot(x) + 11*cot (x) + 33*cot (x) + 66*cot  (x) + 110*cot (x) + 126*cot  (x) + ---- + ---------- + ---- + ----------- + ------------ + ----------- + ------------ + -------------
 | |          24       3  |                                                                                           9         23        3          7             19             7             17              13     
 | \       sin  (x)       /                                                                                                                                                                                            
 |                                                                                                                                                                                                                     
/                                                                                                                                                                                                                      
$$\int \left(\frac{2 x^{2}}{3} + \left(8 x^{5} - \frac{3}{\sin^{24}{\left(x \right)}}\right)\right)\, dx = C + \frac{4 x^{6}}{3} + \frac{2 x^{3}}{9} + \frac{3 \cot^{23}{\left(x \right)}}{23} + \frac{11 \cot^{21}{\left(x \right)}}{7} + \frac{165 \cot^{19}{\left(x \right)}}{19} + \frac{495 \cot^{17}{\left(x \right)}}{17} + 66 \cot^{15}{\left(x \right)} + \frac{1386 \cot^{13}{\left(x \right)}}{13} + 126 \cot^{11}{\left(x \right)} + 110 \cot^{9}{\left(x \right)} + \frac{495 \cot^{7}{\left(x \right)}}{7} + 33 \cot^{5}{\left(x \right)} + 11 \cot^{3}{\left(x \right)} + 3 \cot{\left(x \right)}$$
The graph
The answer [src]
-oo
$$-\infty$$
=
=
-oo
$$-\infty$$
-oo
Numerical answer [src]
-1.95714425086013e+437
-1.95714425086013e+437

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