Grade 12Integral Calculus

integrate the following integral
∫ cos^4 (2x) dx

taniska

11 Years agoGrade 12

2 Answers

Jitender Singh

11 Years ago

Ans:
Hello Student,
Please find answer to your question below

$I = \int cos^{4}2xdx$
$I = \int cos^{2}2x.cos^{2}2xdx$
$I = \int cos^{2}2x(1-sin^{2}2x)dx$
$I = \int cos^{2}2xdx-\int cos^{2}2xsin^{2}2xdx$
$I = \frac{1}{2}\int 2cos^{2}2xdx-\frac{1}{4}\int 4cos^{2}2xsin^{2}2xdx$
$I = \frac{1}{2}\int (1+cos4x)dx-\frac{1}{4}\int (2sin2xcos2x)^2dx$
$I = \frac{1}{2}\int (1+cos4x)dx-\frac{1}{4}\int (sin4x)^2dx$
$I = \frac{1}{2}\int (1+cos4x)dx-\frac{1}{8}\int 2(sin4x)^2dx$
$I = \frac{1}{2}\int (1+cos4x)dx-\frac{1}{8}\int (1-cos8x)dx$
$I = \frac{1}{2}(x+\frac{sin4x}{4})-\frac{1}{8}(x-\frac{sin8x}{8})+c$