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

The integral of (cos(a)x+sin(a)x)^2 is (x^3cos^2(a))/3+1/3 x^3sin(2a)+(x^3sin^2(a))/3+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (cos(a)x+sin(a)x)^2

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

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

Solution steps

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

Expand

(cos (a) x + sin (a) x)^{2} : x^{2} cos^{2} (a) + 2 x^{2} cos (a) sin (a) + x^{2} sin^{2} (a)

= \int x^{2} cos^{2} (a) + 2 x^{2} cos (a) sin (a) + x^{2} sin^{2} (a) 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

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

\int x^{2} cos^{2} (a) d x = \frac{x ^{3} cos ^{2} ( a )}{3}

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

\int x^{2} sin^{2} (a) d x = \frac{x ^{3} sin ^{2} ( a )}{3}

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

Add a constant to the solution

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

\int \frac{sin ( 3 x )}{2} d x

\int (x + 2 x)^{2} d x

\int \frac{2}{20 - x} d x

\int (1 + 2 x)^{4} x d x

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

What is the integral of (cos(a)x+sin(a)x)^2 ?
The integral of (cos(a)x+sin(a)x)^2 is (x^3cos^2(a))/3+1/3 x^3sin(2a)+(x^3sin^2(a))/3+C