Correct Answer is : 141/441 sec x tan⁡3³tan3 x + 585/885 sec x tan x + 383/883 log (sec x + tan x) + c

1/4 sec x ³ x + 5/8 sec x tan x + 3/4 (log sec x + tan x) + c

1/4 sec x ³ x + 5/8 sec x tan x + 3/8 log (sec x + tan x) + c

1/4 ² x tan x + 3/8 sec x tan x + 3/4 log (sec x + tan x) + c

1/4 sec x ³ x + 11/8 sec x tan x + 3/4 log (sec x + tan x) + c

Correct Answer is : 141/441 sec x tan⁡3³tan3 x + 585/885 sec x tan x + 383/883 log (sec x + tan x) + c

Test Index

TS EAMCET 04 May 2019 Shift 1 Solved Paper

Section: Mathematics

Question : 74 of 160

Marks: +1, -0

∫

\sec^{5}

x dx =

$\frac{1}{4}$ sec x $\tan^{3}$ x + $\frac{5}{8}$ sec x tan x + $\frac{3}{4}$ (log sec x + tan x) + c
$\frac{1}{4}$ sec x $\tan^{3}$ x + $\frac{5}{8}$ sec x tan x + $\frac{3}{8}$ log (sec x + tan x) + c
$\frac{1}{4}$ $\sec^{2}$ x tan x + $\frac{3}{8}$ sec x tan x + $\frac{3}{4}$ log (sec x + tan x) + c
$\frac{1}{4}$ sec x $\tan^{3}$ x + $\frac{11}{8}$ sec x tan x + $\frac{3}{4}$ log (sec x + tan x) + c

Solution:

Consider the integral,

I=\int \sec^{5} dx

Simplify the above,

I \;\; =\int \sec^{3} x \sec^{2} x dx

\;\; =\sec^{3} x \tan x - \int 3 \sec^{3} x \tan^{2} x dx

\;\; =\sec^{3} x \tan x - 3 \int \sec^{3} x (1+\sec^{2} x) dx

\;\; =\sec^{3} x \tan x - 3 \int \sec^{5} x dx - 3 \int \sec^{3} x dx

Further simplify the above,

I=\sec^{3} x \tan x - 3 I - 3 \int \sec^{2} x \sqrt{1+\tan^{2} x} dx

Take

\tan x = t

then

4 I \;\; =\sec^{3} x \tan x - 3 \int \sqrt{t^{2}+1} dx

4 I \;\; =\sec^{3} x \tan x - \; \frac{3t}{2} \sqrt{1+t^{2}} + \; \frac{3}{2} \log |t+\sqrt{1+t^{2}}| + c

I \;\; =\; \frac{\sec^{3} x \tan x}{4} - \; \frac{3}{8} \sec x \tan x + \; \frac{3}{8} \log |\sec x+\tan x| + c

\;\; =\; \frac{(\sec^{2} x) \sec x \tan x}{4} + \; \frac{3}{8} \sec x \tan x + \; \frac{3}{8} \log |\sec x+\tan x| + c

Further simplify

I \;\; =\; \frac{(1+\tan^{2} x) \sec x \tan x}{4} + \; \frac{3}{8} \sec x \tan x + \; \frac{3}{8} \log |\sec x+\tan x| + c

\;\; =\; \frac{\sec x \tan^{3} x}{4} + \; \frac{5}{8} \sec x \tan x + \; \frac{3}{8} \log |\sec x+\tan x| + c

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