What is the integral of (sin^7(x))/(cos^3(x)) ?

The integral of (sin^7(x))/(cos^3(x)) is 1/2 sec^2(x)+3ln|cos(x)|-3/2 cos^2(x)+1/4 cos^4(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin^7(x))/(cos^3(x))

\int \frac{sin ^{7} ( x )}{cos ^{3} ( x )} d x

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

Solution steps

\int \frac{sin ^{7} ( x )}{cos ^{3} ( x )} d x

Rewrite using trig identities

= \int \frac{( 1 - cos ^{2} ( x ) ) ^{3} sin ( x )}{cos ^{3} ( x )} d x

Apply u-substitution

: \int - \frac{( 1 - u ^{2} ) ^{3}}{u ^{3}} d u

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

Expand

- \frac{( 1 - u ^{2} ) ^{3}}{u ^{3}} : - \frac{1}{u ^{3}} + \frac{3}{u} - 3 u + u^{3}

= \int - \frac{1}{u ^{3}} + \frac{3}{u} - 3 u + u^{3} d u

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}{u ^{3}} d u + \int \frac{3}{u} d u - \int 3 u d u + \int u^{3} d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{3}{u} d u = 3 ln ∣ u ∣

\int 3 u d u = \frac{3 u ^{2}}{2}

\int u^{3} d u = \frac{u ^{4}}{4}

= - (- \frac{1}{2 u ^{2}}) + 3 ln ∣ u ∣ - \frac{3 u ^{2}}{2} + \frac{u ^{4}}{4}

Substitute back

u = cos (x)

= - (- \frac{1}{2 cos ^{2} ( x )}) + 3 ln ∣ cos (x) ∣ - \frac{3 cos ^{2} ( x )}{2} + \frac{cos ^{4} ( x )}{4}

Simplify

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

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

Add a constant to the solution

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

\int (x^{3} + 3 x) d x

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

\int \frac{1}{( 289 + x ^{2} ) ^{\frac{3}{2}}} d x

\int \frac{26 s + 26}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

\int - 9 x cos (7 x) d x

What is the integral of (sin^7(x))/(cos^3(x)) ?
The integral of (sin^7(x))/(cos^3(x)) is 1/2 sec^2(x)+3ln|cos(x)|-3/2 cos^2(x)+1/4 cos^4(x)+C