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integral of 4/(35sin(x)-12cos(x))

Solution

\int \frac{4}{35 sin ( x ) - 12 cos ( x )} d x

- \frac{4}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37}) + C

Solution steps

\int \frac{4}{35 sin ( x ) - 12 cos ( x )} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{35 sin ( x ) - 12 cos ( x )} d x

Apply u-substitution

: \int \frac{1}{35 u - 6 ( - u ^{2} + 1 )} d u

= 4 \cdot \int \frac{1}{35 u - 6 ( - u ^{2} + 1 )} d u

Complete the square

35 u - 6 (- u^{2} + 1) : 6 (u + \frac{35}{12})^{2} - \frac{1369}{24}

= 4 \cdot \int \frac{1}{6 ( u + \frac{35}{12} ) ^{2} - \frac{1369}{24}} d u

Apply u-substitution

: \int \frac{24}{144 v ^{2} - 1369} d v

= 4 \cdot \int \frac{24}{144 v ^{2} - 1369} d v

Take the constant out:

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

= 4 \cdot 24 \cdot \int \frac{1}{144 v ^{2} - 1369} d v

Apply Integral Substitution

= 4 \cdot 24 \cdot \int \frac{1}{444 ( w ^{2} - 1 )} d w

Take the constant out:

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

= 4 \cdot 24 \cdot \frac{1}{444} \cdot \int \frac{1}{w ^{2} - 1} d w

Factor

w^{2} - 1 : - (- w^{2} + 1)

= 4 \cdot 24 \cdot \frac{1}{444} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

Take the constant out:

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

= 4 \cdot 24 \cdot \frac{1}{444} (- \int \frac{1}{- w ^{2} + 1} d w)

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 4 \cdot 24 \cdot \frac{1}{444} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

Substitute back

= 4 \cdot 24 \cdot \frac{1}{444} (- (\frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) + 1}{2} - \frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) - 1}{2}))

Simplify

4 \cdot 24 \cdot \frac{1}{444} (- (\frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) + 1}{2} - \frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) - 1}{2})) : - \frac{4}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37})

= - \frac{4}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37})

Add a constant to the solution

= - \frac{4}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37}) + C

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