What is the integral of ((1+sin(t))^3)/(cos(t)) ?

The integral of ((1+sin(t))^3)/(cos(t)) is 4ln|tan(t)+sec(t)|-4ln|cos(t)|-3sin(t)+1/2 cos^2(t)+C

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integral of ((1+sin(t))^3)/(cos(t))

\int \frac{( 1 + sin ( t ) ) ^{3}}{cos ( t )} d t

4 ln ∣ tan (t) + sec (t) ∣ - 4 ln ∣ cos (t) ∣ - 3 sin (t) + \frac{1}{2} cos^{2} (t) + C

Solution steps

\int \frac{( 1 + sin ( t ) ) ^{3}}{cos ( t )} d t

Expand

\frac{( 1 + sin ( t ) ) ^{3}}{cos ( t )} : \frac{1}{cos ( t )} + \frac{3 sin ( t )}{cos ( t )} + \frac{3 sin ^{2} ( t )}{cos ( t )} + \frac{sin ^{3} ( t )}{cos ( t )}

= \int \frac{1}{cos ( t )} + \frac{3 sin ( t )}{cos ( t )} + \frac{3 sin ^{2} ( t )}{cos ( t )} + \frac{sin ^{3} ( t )}{cos ( t )} d t

Apply the Sum Rule:

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

= \int \frac{1}{cos ( t )} d t + \int \frac{3 sin ( t )}{cos ( t )} d t + \int \frac{3 sin ^{2} ( t )}{cos ( t )} d t + \int \frac{sin ^{3} ( t )}{cos ( t )} d t

\int \frac{1}{cos ( t )} d t = ln ∣ tan (t) + sec (t) ∣

\int \frac{3 sin ( t )}{cos ( t )} d t = - 3 ln ∣ cos (t) ∣

\int \frac{3 sin ^{2} ( t )}{cos ( t )} d t = 3 (ln ∣ tan (t) + sec (t) ∣ - sin (t))

\int \frac{sin ^{3} ( t )}{cos ( t )} d t = - ln ∣ cos (t) ∣ + \frac{cos ^{2} ( t )}{2}

= ln ∣ tan (t) + sec (t) ∣ - 3 ln ∣ cos (t) ∣ + 3 (ln ∣ tan (t) + sec (t) ∣ - sin (t)) - ln ∣ cos (t) ∣ + \frac{cos ^{2} ( t )}{2}

Simplify

ln ∣ tan (t) + sec (t) ∣ - 3 ln ∣ cos (t) ∣ + 3 (ln ∣ tan (t) + sec (t) ∣ - sin (t)) - ln ∣ cos (t) ∣ + \frac{cos ^{2} ( t )}{2} : 4 ln ∣ tan (t) + sec (t) ∣ - 4 ln ∣ cos (t) ∣ - 3 sin (t) + \frac{1}{2} cos^{2} (t)

= 4 ln ∣ tan (t) + sec (t) ∣ - 4 ln ∣ cos (t) ∣ - 3 sin (t) + \frac{1}{2} cos^{2} (t)

Add a constant to the solution

= 4 ln ∣ tan (t) + sec (t) ∣ - 4 ln ∣ cos (t) ∣ - 3 sin (t) + \frac{1}{2} cos^{2} (t) + C

\int \frac{1}{x + 3 - x + 2} d x

\int \frac{2}{x ^{2} + 4 x + 5} d x

\int (\frac{1}{x ^{2}} + \frac{5}{x} - 1) d x

\int - 6 x^{6} - 4 x^{5} + 5 x - 3 d x

\int (16 x + 5) d x

What is the integral of ((1+sin(t))^3)/(cos(t)) ?
The integral of ((1+sin(t))^3)/(cos(t)) is 4ln|tan(t)+sec(t)|-4ln|cos(t)|-3sin(t)+1/2 cos^2(t)+C