What is $F (x) = \int sin 2 x {cos}^{2} x - sin x {cos}^{2} x d x$ $F(x) = int sin2xcos^2x-sinxcos^2x dx$ if $F (π) = 1$ $F(pi) = 1$ ?

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What is $F (x) = \int sin 2 x {cos}^{2} x - sin x {cos}^{2} x d x$ $F(x) = int sin2xcos^2x-sinxcos^2x dx$ if $F (π) = 1$ $F(pi) = 1$ ?

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Answer 1 · 2017-01-09T14:05:55

$F (x) = \frac{{cos}^{3} x}{3} - \frac{{cos}^{4} x}{2} + \frac{11}{6}$ $F(x) = (cos^3x)/3- (cos^4x)/2+11/6$

Explanation:

Let's calculate the indefinite integral:

$\int (sin 2 x {cos}^{2} x - sin x {cos}^{2} x) d x = \int {cos}^{2} x (sin 2 x - sin x) d x$ $int (sin2xcos^2x-sinxcos^2x)dx = int cos^2x(sin2x-sinx)dx$

Using the identity:

$sin 2 x = 2 sin x cos x$ $sin2x = 2 sinxcosx$

$\int {cos}^{2} x (sin 2 x - sin x) d x = \int {cos}^{2} x (2 sin x cos x - sin x) d x = \int {cos}^{2} x sin x (2 cos x - 1) d x$ $int cos^2x(sin2x-sinx)dx = int cos^2x(2sinxcosx-sinx)dx=int cos^2xsinx(2cosx-1)dx$

As $sin x d x = - d (cos x)$ $sinxdx = -d(cosx)$

$\int {cos}^{2} x sin x (2 cos x - 1) d x = \int ({cos}^{2} x - 2 {cos}^{3} x) d (cos x) = \frac{{cos}^{3} x}{3} - \frac{{cos}^{4} x}{2} + C$ $int cos^2xsinx(2cosx-1)dx = int (cos^2x-2cos^3x)d(cosx) = (cos^3x)/3- (cos^4x)/2+C$

Now we determine the constant from:

$F (π) = 1$ $F(pi) = 1$

$\frac{{cos}^{3} π}{3} - \frac{{cos}^{4} π}{2} + C = 1$ $(cos^3pi)/3- (cos^4pi)/2+C = 1$

$\frac{{(- 1)}^{3}}{3} - \frac{{(- 1)}^{4}}{2} + C = 1$ $(-1)^3/3-(-1)^4/2 +C = 1$

$- \frac{1}{3} - \frac{1}{2} + C = 1$ $-1/3-1/2 +C =1$

$C = 1 + \frac{5}{6} = \frac{11}{6}$ $C = 1+5/6=11/6$

So, finally:

$F (x) = \frac{{cos}^{3} π}{3} - \frac{{cos}^{4} π}{2} + \frac{11}{6}$ $F(x) = (cos^3pi)/3- (cos^4pi)/2+11/6$

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What is $F (x) = \int sin 2 x {cos}^{2} x - sin x {cos}^{2} x d x$ $F(x) = int sin2xcos^2x-sinxcos^2x dx$ if $F (π) = 1$ $F(pi) = 1$ ?

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Explanation:

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What is F(x) = int sin2xcos^2x-sinxcos^2x dxF(x)=∫sin2xcos2x−sinxcos2xdxF(x) = int sin2xcos^2x-sinxcos^2x dx if F(pi) = 1 F(π)=1F(pi) = 1 ?

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Explanation:

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What is $F (x) = \int sin 2 x {cos}^{2} x - sin x {cos}^{2} x d x$ $F(x) = int sin2xcos^2x-sinxcos^2x dx$ if $F (π) = 1$ $F(pi) = 1$ ?