What is the integral of (1+cos^2(5)x^2csc^2(5)x^2)6x ?

The integral of (1+cos^2(5)x^2csc^2(5)x^2)6x is 6csc^2(5)((sin^2(5)x^2)/2+(cos^2(5)x^6)/6)+C

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integral of (1+cos^2(5)x^2csc^2(5)x^2)6x

\int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) 6 x d x

6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6}) + C

Solution steps

\int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) \cdot 6 x d x

Take the constant out:

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

= 6 \cdot \int (1 + cos^{2} (5) x^{2} csc^{2} (5) x^{2}) x d x

Rewrite using trig identities

= 6 \cdot \int \frac{x ( sin ^{2} ( 5 ) + cos ^{2} ( 5 ) x ^{4} )}{sin ^{2} ( 5 )} d x

Take the constant out:

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

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} \cdot \int x (sin^{2} (5) + cos^{2} (5) x^{4}) d x

Expand

x (sin^{2} (5) + cos^{2} (5) x^{4}) : sin^{2} (5) x + cos^{2} (5) x^{5}

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} \cdot \int sin^{2} (5) x + cos^{2} (5) x^{5} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} (\int sin^{2} (5) x d x + \int cos^{2} (5) x^{5} d x)

\int sin^{2} (5) x d x = \frac{sin ^{2} ( 5 ) x ^{2}}{2}

\int cos^{2} (5) x^{5} d x = \frac{cos ^{2} ( 5 ) x ^{6}}{6}

= 6 \cdot \frac{1}{sin ^{2} ( 5 )} (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6})

Simplify

6 \cdot \frac{1}{sin ^{2} ( 5 )} (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6}) : 6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6})

= 6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6})

Add a constant to the solution

= 6 csc^{2} (5) (\frac{sin ^{2} ( 5 ) x ^{2}}{2} + \frac{cos ^{2} ( 5 ) x ^{6}}{6}) + C

What is the integral of (1+cos^2(5)x^2csc^2(5)x^2)6x ?
The integral of (1+cos^2(5)x^2csc^2(5)x^2)6x is 6csc^2(5)((sin^2(5)x^2)/2+(cos^2(5)x^6)/6)+C