Question

Evaluate: $\int se{c}^{4/3}xcose{c}^{8/3}xdx$

• A. $\frac{3}{5}ta{n}^{-5/3}x-3ta{n}^{1/3}x+C$
• B. $-\frac{3}{5}ta{n}^{-5/3}x-3ta{n}^{1/3}x+C$
• C. $-\frac{3}{5}ta{n}^{-5/3}x-3ta{n}^{1/3}x+C$
• D. None of these

Answer 1

Answer: (B)

Let $I=\int se{c}^{4/3}xcose{c}^{8/3}xdx$

$\Rightarrow I=\int co{s}^{-4/3}xsi{n}^{-8/3}xdx$

$\Rightarrow I=\int \frac{se{c}^{4}x}{ta{n}^{\frac{8}{3}}x}dx=\int \frac{\left(1+ta{n}^{2}x\right)}{ta{n}^{\frac{8}{3}}x}se{c}^{2}xdx$

Putting $tanx=t\Rightarrow se{c}^{2}xdx=dt$

$I=\int \frac{1+t^2}{t^{\frac{8}{3}}}dt=\int \left(t^{-\frac{8}{3}}+t^{-\frac{2}{3}}\right)dt$

$\Rightarrow I=\frac{-3}{5}{t}^{-\frac{5}{3}}+3t^{\frac{1}{3}}+C$

$=\frac{-3}{5}ta{n}^{-5/3}x+3ta{n}^{1/3}x+C$