How do you find the integral of ${sin}^{2} (x) {cos}^{4} (x)$ $sin^2(x)cos^4(x)$ ?

Question

How do you find the integral of ${sin}^{2} (x) {cos}^{4} (x)$ $sin^2(x)cos^4(x)$ ?

1 Answer

Impact of this question

2237 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-14T00:03:12

$\int {sin}^{2} (x) {cos}^{4} (x) d x$ $int sin^2(x)cos^4(x) dx$

$= - \frac{1}{6} {sin}^{5} (x) cos (x) + \frac{7}{24} {sin}^{3} (x) cos (x) - \frac{1}{16} sin (x) cos (x) + \frac{1}{16} x + C$ $= -1/6 sin^5(x) cos(x)+7/24 sin^3(x) cos(x)-1/16 sin(x) cos(x)+1/16 x + C$

Explanation:

First note that:

${sin}^{2} (x) {cos}^{4} (x) = (1 - {cos}^{2} (x)) {cos}^{4} (x) = - {cos}^{6} x + {cos}^{4} (x)$ $sin^2(x)cos^4(x) = (1 - cos^2(x)) cos^4(x) = -cos^(6)x + cos^4(x)$

$\frac{d}{d x} {sin}^{5} (x) cos (x) = 5 {sin}^{4} (x) {cos}^{2} (x) - {sin}^{6} (x)$ $d/(dx) sin^5(x) cos(x) = 5 sin^4(x) cos^2(x) - sin^6(x)$

$\frac{d}{d x} {sin}^{5} (x) cos (x) = {sin}^{4} (x) (5 {cos}^{2} (x) - {sin}^{2} (x))$ $color(white)(d/(dx) sin^5(x) cos(x)) = sin^4(x)(5 cos^2(x) - sin^2(x))$

$\frac{d}{d x} {sin}^{5} (x) cos (x) = (1 - {cos}^{2} (x)) (1 - {cos}^{2} (x)) (5 {cos}^{2} (x) - (1 - {cos}^{2} (x)))$ $color(white)(d/(dx) sin^5(x) cos(x)) = (1-cos^2(x))(1-cos^2(x))(5 cos^2(x) - (1-cos^2(x)))$

$\frac{d}{d x} {sin}^{5} (x) cos (x) = ({cos}^{4} (x) - 2 {cos}^{2} (x) + 1) (6 {cos}^{2} (x) - 1)$ $color(white)(d/(dx) sin^5(x) cos(x)) = (cos^4(x)-2cos^2(x)+1)(6 cos^2(x) - 1)$

$\frac{d}{d x} {sin}^{5} (x) cos (x) = 6 {cos}^{6} (x) - 13 {cos}^{4} (x) + 8 {cos}^{2} (x) - 1$ $color(white)(d/(dx) sin^5(x) cos(x)) = 6cos^6(x)-13cos^4(x)+8cos^2(x)-1$

$\frac{d}{d x} {sin}^{3} (x) cos (x) = 3 {sin}^{2} (x) {cos}^{2} (x) - {sin}^{4} (x)$ $d/(dx) sin^3(x) cos(x) = 3 sin^2(x) cos^(2)(x) - sin^4(x)$

$\frac{d}{d x} {sin}^{3} (x) cos (x) = {sin}^{2} (x) (3 {cos}^{2} (x) - {sin}^{2} (x))$ $color(white)(d/(dx) sin^3(x) cos(x)) = sin^2(x)(3 cos^2(x) - sin^2(x))$

$\frac{d}{d x} {sin}^{3} (x) cos (x) = (1 - {cos}^{2} (x)) (3 {cos}^{2} (x) - (1 - {cos}^{2} (x)))$ $color(white)(d/(dx) sin^3(x) cos(x)) = (1-cos^2(x))(3 cos^2(x) - (1-cos^2(x)))$

$\frac{d}{d x} {sin}^{3} (x) cos (x) = (- {cos}^{2} (x) + 1) (4 {cos}^{2} (x) - 1)$ $color(white)(d/(dx) sin^3(x) cos(x)) = (-cos^2(x)+1)(4 cos^2(x) - 1)$

$\frac{d}{d x} {sin}^{3} (x) cos (x) = - 4 {cos}^{4} (x) + 5 {cos}^{2} (x) - 1$ $color(white)(d/(dx) sin^3(x) cos(x)) = -4cos^4(x)+5cos^2(x)-1$

$\frac{d}{d x} sin (x) cos (x) = {cos}^{2} (x) - {sin}^{2} (x) = 2 {cos}^{2} (x) - 1$ $d/(dx) sin(x) cos(x) = cos^2(x)-sin^2(x) = 2cos^2(x) - 1$

Now we can choose multipliers to make the running sum match each coefficient of ${cos}^{2 k} (x)$ $cos^(2k)(x)$ in descending order:

$\frac{d}{d x} (- \frac{1}{6} {sin}^{5} (x) cos (x)) = - {cos}^{6} (x) + \frac{13}{6} {cos}^{4} (x) - \frac{4}{3} {cos}^{2} (x) + \frac{1}{6}$ $d/(dx) (-1/6 sin^5(x) cos(x)) = -cos^6(x)+13/6 cos^4(x)-4/3 cos^2(x)+1/6$

$\frac{d}{d x} (\frac{7}{24} {sin}^{3} (x) cos (x)) = - \frac{7}{6} {cos}^{4} (x) + \frac{35}{24} {cos}^{2} (x) - \frac{7}{24}$ $d/(dx) (7/24 sin^3(x) cos(x)) = -7/6 cos^4(x)+35/24 cos^2(x) - 7/24$

$\frac{d}{d x} (- \frac{1}{16} sin (x) cos (x)) = - \frac{1}{8} {cos}^{2} (x) + \frac{1}{16}$ $d/(dx) (-1/16 sin(x) cos(x)) = -1/8 cos^2(x) + 1/16$

$\frac{d}{d x} (\frac{1}{16} x) = \frac{1}{16}$ $d/(dx) (1/16 x) = 1/16$

summing to: $- {cos}^{6} (x) + {cos}^{4} (x) = {sin}^{2} (x) {cos}^{4} (x)$ $-cos^6(x)+cos^4(x) = sin^2(x)cos^4(x)$

So:

$\int {sin}^{2} (x) {cos}^{4} (x) d x$ $int sin^2(x)cos^4(x) dx$

$= - \frac{1}{6} {sin}^{5} (x) cos (x) + \frac{7}{24} {sin}^{3} (x) cos (x) - \frac{1}{16} sin (x) cos (x) + \frac{1}{16} x + C$ $= -1/6 sin^5(x) cos(x)+7/24 sin^3(x) cos(x)-1/16 sin(x) cos(x)+1/16 x + C$

Science

Math

Humanities

... and beyond

How do you find the integral of ${sin}^{2} (x) {cos}^{4} (x)$ $sin^2(x)cos^4(x)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you find the integral of sin2(x)cos4(x)sin^2(x)cos^4(x) ?

1 Answer

Explanation:

Related questions

Impact of this question

How do you find the integral of ${sin}^{2} (x) {cos}^{4} (x)$ $sin^2(x)cos^4(x)$ ?