What is the integral of cos(x)sin(x)sqrt(1+cos^2(x)) ?

The integral of cos(x)sin(x)sqrt(1+cos^2(x)) is -1/3 (1+cos^2(x))^{3/2}+C

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integral of cos(x)sin(x)sqrt(1+cos^2(x))

\int cos (x) sin (x) 1 + cos^{2} (x) d x

- \frac{1}{3} (1 + cos^{2} (x))^{\frac{3}{2}} + C

Solution steps

\int cos (x) sin (x) 1 + cos^{2} (x) d x

Rewrite using trig identities

= \int \frac{1}{2} sin (2 x) 1 + cos^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sin (2 x) 1 + cos^{2} (x) d x

Apply u-substitution

= \frac{1}{2} \cdot \int - 2 u^{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} (- 2 \cdot \int u^{2} d u)

Apply the Power Rule

= \frac{1}{2} (- 2 \cdot \frac{u ^{3}}{3})

Substitute back

u = 1 + cos^{2} (x)

= \frac{1}{2} - 2 \cdot \frac{( 1 + cos ^{2} ( x ) ) ^{3}}{3}

Simplify

\frac{1}{2} - 2 \cdot \frac{( 1 + cos ^{2} ( x ) ) ^{3}}{3} : - \frac{1}{3} (1 + cos^{2} (x))^{\frac{3}{2}}

= - \frac{1}{3} (1 + cos^{2} (x))^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{3} (1 + cos^{2} (x))^{\frac{3}{2}} + C

What is the integral of cos(x)sin(x)sqrt(1+cos^2(x)) ?
The integral of cos(x)sin(x)sqrt(1+cos^2(x)) is -1/3 (1+cos^2(x))^{3/2}+C