evaluate the integral.\n int 3\tan xsec^{7}x dx\n int 3\tan xsec^{7}x dx=square
evaluate the integral.\n int 3\tan xsec^{7}x dx\n int 3\tan xsec^{7}x dx=square
Answer
Explanation:
Step1: Use substitution
Let (u = \sec x), then (du=\sec x\tan xdx). The integral (\int3\tan x\sec^{7}x dx = 3\int\sec^{6}x\cdot\sec x\tan xdx). After substitution, it becomes (3\int u^{6}du).
Step2: Integrate using power - rule
The power - rule for integration is (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)). For (3\int u^{6}du), we have (3\times\frac{u^{7}}{7}+C=\frac{3}{7}u^{7}+C).
Step3: Substitute back
Substitute (u = \sec x) back into the result. So the integral (\int3\tan x\sec^{7}x dx=\frac{3}{7}\sec^{7}x + C).
Answer:
(\frac{3}{7}\sec^{7}x + C)