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Question Number 57572 by cesar.marval.larez@gmail.com last updated on 07/Apr/19

$$\int {\sec }^{4}2\mathrm{xdx}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 07/Apr/19

$$\int {sec}^{2}2x\times {sec}^{2}2xdx$$$$\int (1+{tan}^{2}2x){sec}^{2}2xdx$$$$t=tan2x$$$$\frac{dt}{dx}={sec}^{2}2x\times 2$$$${sec}^{2}2xdx=\frac{dt}{2}$$$$\int (1+t^2)\times \frac{dt}{2}$$$$\frac{1}{2}[\int dt+\int t^{2}dt]$$$$\frac{1}{2}(t+\frac{t^3}{3})+c$$$$\frac{1}{2}(tan2x+\frac{{tan}^{3}2x}{3})+c$$$$$$$$$$

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