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tgx^7sec^4x
Solving integrals/ tgx^7sec^4x

Integral of tgx^7sec^4x dx

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 |  tan (x)*sec (x) dx
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∫01tan⁡7(x)sec⁡4(x) dx\int\limits_{0}^{1} \tan^{7}{\left(x \right)} \sec^{4}{\left(x \right)}\, dx 
Integral(tan(x)^7*sec(x)^4, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    tan⁡7(x)sec⁡4(x)=(sec⁡2(x)−1)3tan⁡(x)sec⁡4(x)\tan^{7}{\left(x \right)} \sec^{4}{\left(x \right)} = \left(\sec^{2}{\left(x \right)} - 1\right)^{3} \tan{\left(x \right)} \sec^{4}{\left(x \right)}

  2. There are multiple ways to do this integral.

    Method #1

    1. Let u=sec⁡2(x)u = \sec^{2}{\left(x \right)}.

      Then let du=2tan⁡(x)sec⁡2(x)dxdu = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx and substitute dudu:

      ∫(u42−3u32+3u22−u2) du\int \left(\frac{u^{4}}{2} - \frac{3 u^{3}}{2} + \frac{3 u^{2}}{2} - \frac{u}{2}\right)\, du

      1. Integrate term-by-term:

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

          ∫u42 du=∫u4 du2\int \frac{u^{4}}{2}\, du = \frac{\int u^{4}\, du}{2}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          So, the result is: u510\frac{u^{5}}{10}

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

          ∫(−3u32) du=−3∫u3 du2\int \left(- \frac{3 u^{3}}{2}\right)\, du = - \frac{3 \int u^{3}\, du}{2}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

          So, the result is: −3u48- \frac{3 u^{4}}{8}

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

          ∫3u22 du=3∫u2 du2\int \frac{3 u^{2}}{2}\, du = \frac{3 \int u^{2}\, du}{2}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

          So, the result is: u32\frac{u^{3}}{2}

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

          ∫(−u2) du=−∫u du2\int \left(- \frac{u}{2}\right)\, du = - \frac{\int u\, du}{2}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u du=u22\int u\, du = \frac{u^{2}}{2}

          So, the result is: −u24- \frac{u^{2}}{4}

        The result is: u510−3u48+u32−u24\frac{u^{5}}{10} - \frac{3 u^{4}}{8} + \frac{u^{3}}{2} - \frac{u^{2}}{4}

      Now substitute uu back in:

      sec⁡10(x)10−3sec⁡8(x)8+sec⁡6(x)2−sec⁡4(x)4\frac{\sec^{10}{\left(x \right)}}{10} - \frac{3 \sec^{8}{\left(x \right)}}{8} + \frac{\sec^{6}{\left(x \right)}}{2} - \frac{\sec^{4}{\left(x \right)}}{4}

    Method #2

    1. Rewrite the integrand:

      (sec⁡2(x)−1)3tan⁡(x)sec⁡4(x)=tan⁡(x)sec⁡10(x)−3tan⁡(x)sec⁡8(x)+3tan⁡(x)sec⁡6(x)−tan⁡(x)sec⁡4(x)\left(\sec^{2}{\left(x \right)} - 1\right)^{3} \tan{\left(x \right)} \sec^{4}{\left(x \right)} = \tan{\left(x \right)} \sec^{10}{\left(x \right)} - 3 \tan{\left(x \right)} \sec^{8}{\left(x \right)} + 3 \tan{\left(x \right)} \sec^{6}{\left(x \right)} - \tan{\left(x \right)} \sec^{4}{\left(x \right)}

    2. Integrate term-by-term:

      1. Let u=sec⁡10(x)u = \sec^{10}{\left(x \right)}.

        Then let du=10tan⁡(x)sec⁡10(x)dxdu = 10 \tan{\left(x \right)} \sec^{10}{\left(x \right)} dx and substitute du10\frac{du}{10}:

        ∫1100 du\int \frac{1}{100}\, du

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

          ∫110 du=∫1 du10\int \frac{1}{10}\, du = \frac{\int 1\, du}{10}

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

            ∫1 du=u\int 1\, du = u

          So, the result is: u10\frac{u}{10}

        Now substitute uu back in:

        sec⁡10(x)10\frac{\sec^{10}{\left(x \right)}}{10}

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

        ∫(−3tan⁡(x)sec⁡8(x)) dx=−3∫tan⁡(x)sec⁡8(x) dx\int \left(- 3 \tan{\left(x \right)} \sec^{8}{\left(x \right)}\right)\, dx = - 3 \int \tan{\left(x \right)} \sec^{8}{\left(x \right)}\, dx

        1. Let u=sec⁡8(x)u = \sec^{8}{\left(x \right)}.

          Then let du=8tan⁡(x)sec⁡8(x)dxdu = 8 \tan{\left(x \right)} \sec^{8}{\left(x \right)} dx and substitute du8\frac{du}{8}:

          ∫164 du\int \frac{1}{64}\, du

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

            ∫18 du=∫1 du8\int \frac{1}{8}\, du = \frac{\int 1\, du}{8}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u8\frac{u}{8}

          Now substitute uu back in:

          sec⁡8(x)8\frac{\sec^{8}{\left(x \right)}}{8}

        So, the result is: −3sec⁡8(x)8- \frac{3 \sec^{8}{\left(x \right)}}{8}

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

        ∫3tan⁡(x)sec⁡6(x) dx=3∫tan⁡(x)sec⁡6(x) dx\int 3 \tan{\left(x \right)} \sec^{6}{\left(x \right)}\, dx = 3 \int \tan{\left(x \right)} \sec^{6}{\left(x \right)}\, dx

        1. Let u=sec⁡6(x)u = \sec^{6}{\left(x \right)}.

          Then let du=6tan⁡(x)sec⁡6(x)dxdu = 6 \tan{\left(x \right)} \sec^{6}{\left(x \right)} dx and substitute du6\frac{du}{6}:

          ∫136 du\int \frac{1}{36}\, du

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

            ∫16 du=∫1 du6\int \frac{1}{6}\, du = \frac{\int 1\, du}{6}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u6\frac{u}{6}

          Now substitute uu back in:

          sec⁡6(x)6\frac{\sec^{6}{\left(x \right)}}{6}

        So, the result is: sec⁡6(x)2\frac{\sec^{6}{\left(x \right)}}{2}

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

        ∫(−tan⁡(x)sec⁡4(x)) dx=−∫tan⁡(x)sec⁡4(x) dx\int \left(- \tan{\left(x \right)} \sec^{4}{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)} \sec^{4}{\left(x \right)}\, dx

        1. Let u=sec⁡4(x)u = \sec^{4}{\left(x \right)}.

          Then let du=4tan⁡(x)sec⁡4(x)dxdu = 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)} dx and substitute du4\frac{du}{4}:

          ∫116 du\int \frac{1}{16}\, du

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

            ∫14 du=∫1 du4\int \frac{1}{4}\, du = \frac{\int 1\, du}{4}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u4\frac{u}{4}

          Now substitute uu back in:

          sec⁡4(x)4\frac{\sec^{4}{\left(x \right)}}{4}

        So, the result is: −sec⁡4(x)4- \frac{\sec^{4}{\left(x \right)}}{4}

      The result is: sec⁡10(x)10−3sec⁡8(x)8+sec⁡6(x)2−sec⁡4(x)4\frac{\sec^{10}{\left(x \right)}}{10} - \frac{3 \sec^{8}{\left(x \right)}}{8} + \frac{\sec^{6}{\left(x \right)}}{2} - \frac{\sec^{4}{\left(x \right)}}{4}

    Method #3

    1. Rewrite the integrand:

      (sec⁡2(x)−1)3tan⁡(x)sec⁡4(x)=tan⁡(x)sec⁡10(x)−3tan⁡(x)sec⁡8(x)+3tan⁡(x)sec⁡6(x)−tan⁡(x)sec⁡4(x)\left(\sec^{2}{\left(x \right)} - 1\right)^{3} \tan{\left(x \right)} \sec^{4}{\left(x \right)} = \tan{\left(x \right)} \sec^{10}{\left(x \right)} - 3 \tan{\left(x \right)} \sec^{8}{\left(x \right)} + 3 \tan{\left(x \right)} \sec^{6}{\left(x \right)} - \tan{\left(x \right)} \sec^{4}{\left(x \right)}

    2. Integrate term-by-term:

      1. Let u=sec⁡10(x)u = \sec^{10}{\left(x \right)}.

        Then let du=10tan⁡(x)sec⁡10(x)dxdu = 10 \tan{\left(x \right)} \sec^{10}{\left(x \right)} dx and substitute du10\frac{du}{10}:

        ∫1100 du\int \frac{1}{100}\, du

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

          ∫110 du=∫1 du10\int \frac{1}{10}\, du = \frac{\int 1\, du}{10}

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

            ∫1 du=u\int 1\, du = u

          So, the result is: u10\frac{u}{10}

        Now substitute uu back in:

        sec⁡10(x)10\frac{\sec^{10}{\left(x \right)}}{10}

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

        ∫(−3tan⁡(x)sec⁡8(x)) dx=−3∫tan⁡(x)sec⁡8(x) dx\int \left(- 3 \tan{\left(x \right)} \sec^{8}{\left(x \right)}\right)\, dx = - 3 \int \tan{\left(x \right)} \sec^{8}{\left(x \right)}\, dx

        1. Let u=sec⁡8(x)u = \sec^{8}{\left(x \right)}.

          Then let du=8tan⁡(x)sec⁡8(x)dxdu = 8 \tan{\left(x \right)} \sec^{8}{\left(x \right)} dx and substitute du8\frac{du}{8}:

          ∫164 du\int \frac{1}{64}\, du

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

            ∫18 du=∫1 du8\int \frac{1}{8}\, du = \frac{\int 1\, du}{8}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u8\frac{u}{8}

          Now substitute uu back in:

          sec⁡8(x)8\frac{\sec^{8}{\left(x \right)}}{8}

        So, the result is: −3sec⁡8(x)8- \frac{3 \sec^{8}{\left(x \right)}}{8}

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

        ∫3tan⁡(x)sec⁡6(x) dx=3∫tan⁡(x)sec⁡6(x) dx\int 3 \tan{\left(x \right)} \sec^{6}{\left(x \right)}\, dx = 3 \int \tan{\left(x \right)} \sec^{6}{\left(x \right)}\, dx

        1. Let u=sec⁡6(x)u = \sec^{6}{\left(x \right)}.

          Then let du=6tan⁡(x)sec⁡6(x)dxdu = 6 \tan{\left(x \right)} \sec^{6}{\left(x \right)} dx and substitute du6\frac{du}{6}:

          ∫136 du\int \frac{1}{36}\, du

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

            ∫16 du=∫1 du6\int \frac{1}{6}\, du = \frac{\int 1\, du}{6}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u6\frac{u}{6}

          Now substitute uu back in:

          sec⁡6(x)6\frac{\sec^{6}{\left(x \right)}}{6}

        So, the result is: sec⁡6(x)2\frac{\sec^{6}{\left(x \right)}}{2}

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

        ∫(−tan⁡(x)sec⁡4(x)) dx=−∫tan⁡(x)sec⁡4(x) dx\int \left(- \tan{\left(x \right)} \sec^{4}{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)} \sec^{4}{\left(x \right)}\, dx

        1. Let u=sec⁡4(x)u = \sec^{4}{\left(x \right)}.

          Then let du=4tan⁡(x)sec⁡4(x)dxdu = 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)} dx and substitute du4\frac{du}{4}:

          ∫116 du\int \frac{1}{16}\, du

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

            ∫14 du=∫1 du4\int \frac{1}{4}\, du = \frac{\int 1\, du}{4}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u4\frac{u}{4}

          Now substitute uu back in:

          sec⁡4(x)4\frac{\sec^{4}{\left(x \right)}}{4}

        So, the result is: −sec⁡4(x)4- \frac{\sec^{4}{\left(x \right)}}{4}

      The result is: sec⁡10(x)10−3sec⁡8(x)8+sec⁡6(x)2−sec⁡4(x)4\frac{\sec^{10}{\left(x \right)}}{10} - \frac{3 \sec^{8}{\left(x \right)}}{8} + \frac{\sec^{6}{\left(x \right)}}{2} - \frac{\sec^{4}{\left(x \right)}}{4}

  3. Now simplify:

    (4sec⁡6(x)−15sec⁡4(x)+20sec⁡2(x)−10)sec⁡4(x)40\frac{\left(4 \sec^{6}{\left(x \right)} - 15 \sec^{4}{\left(x \right)} + 20 \sec^{2}{\left(x \right)} - 10\right) \sec^{4}{\left(x \right)}}{40}

  4. Add the constant of integration:

    (4sec⁡6(x)−15sec⁡4(x)+20sec⁡2(x)−10)sec⁡4(x)40+constant\frac{\left(4 \sec^{6}{\left(x \right)} - 15 \sec^{4}{\left(x \right)} + 20 \sec^{2}{\left(x \right)} - 10\right) \sec^{4}{\left(x \right)}}{40}+ \mathrm{constant}

The answer is:

(4sec⁡6(x)−15sec⁡4(x)+20sec⁡2(x)−10)sec⁡4(x)40+constant\frac{\left(4 \sec^{6}{\left(x \right)} - 15 \sec^{4}{\left(x \right)} + 20 \sec^{2}{\left(x \right)} - 10\right) \sec^{4}{\left(x \right)}}{40}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                 
 |                             6           8         4         10   
 |    7       4             sec (x)   3*sec (x)   sec (x)   sec  (x)
 | tan (x)*sec (x) dx = C + ------- - --------- - ------- + --------
 |                             2          8          4         10   
/
∫tan⁡7(x)sec⁡4(x) dx=C+sec⁡10(x)10−3sec⁡8(x)8+sec⁡6(x)2−sec⁡4(x)4\int \tan^{7}{\left(x \right)} \sec^{4}{\left(x \right)}\, dx = C + \frac{\sec^{10}{\left(x \right)}}{10} - \frac{3 \sec^{8}{\left(x \right)}}{8} + \frac{\sec^{6}{\left(x \right)}}{2} - \frac{\sec^{4}{\left(x \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90500-250
Plot the graph f(x)
The answer [src] 
               2            6            4   
1    4 - 15*cos (1) - 10*cos (1) + 20*cos (1)
-- + ----------------------------------------
40                       10                  
                   40*cos  (1)
140+−15cos⁡2(1)−10cos⁡6(1)+20cos⁡4(1)+440cos⁡10(1)\frac{1}{40} + \frac{- 15 \cos^{2}{\left(1 \right)} - 10 \cos^{6}{\left(1 \right)} + 20 \cos^{4}{\left(1 \right)} + 4}{40 \cos^{10}{\left(1 \right)}} 
=
= 
               2            6            4   
1    4 - 15*cos (1) - 10*cos (1) + 20*cos (1)
-- + ----------------------------------------
40                       10                  
                   40*cos  (1)
140+−15cos⁡2(1)−10cos⁡6(1)+20cos⁡4(1)+440cos⁡10(1)\frac{1}{40} + \frac{- 15 \cos^{2}{\left(1 \right)} - 10 \cos^{6}{\left(1 \right)} + 20 \cos^{4}{\left(1 \right)} + 4}{40 \cos^{10}{\left(1 \right)}}
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Numerical answer [src] 
12.7214684874328
12.7214684874328
The graph
Integral of tgx^7sec^4x dx

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

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