Please calculate 1). ∫sin^5 x dx 2). ∫cos^2 x dx 3). ∫tan^3 x.sec^5 x dx 4). ∫cos 5x.cos 3x dx 5). ∫ ((tan^2 x−1)/(sec^2 x)) dx


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Question Number 156009 by zainaltanjung last updated on 07/Oct/21

$Please calculate$ $1) . \int \sin^{5} x dx$ $2) . \int \cos^{2} x dx$ $3) . \int \tan^{3} x . \sec^{5} x dx$ $4) . \int \cos 5 x . \cos 3 x dx$ $5) . \int \frac{\tan^{2} x - 1}{\sec^{2} x} dx$
Commented by aliyn last updated on 07/Oct/21

$5) \int \frac{{t a n}^{2} x - 1}{{s e c}^{2} x} d x = \int \frac{{s e c}^{2} x - 2}{{s e c}^{2} x} d x$ $= x - \int (1 + c o s 2 x) d x = x - x - \frac{1}{2} s i n 2 x + c$ $= - s i n x c o s x + c$ $⊛ M$
Commented by aliyn last updated on 07/Oct/21

$2) \int {c o s}^{2} x d x = \int \frac{1}{2} (1 + c o s 2 x) d x$ $= \frac{1}{2} x + \frac{1}{2} s i n x c o s x + c$ $⊛ M$
Commented by aliyn last updated on 07/Oct/21

$3) \int {t a n}^{3} x . {s e c}^{5} x d x = \int t a n x s e c x {t a n}^{2} x {s e c}^{4} x d x$ $= \int ({s e c}^{2} x - 1) . {s e c}^{4} x . s e c x . t a n x$ $= \int ({s e c}^{6} x - {s e c}^{4} x) d (s e c x)$ $= \frac{1}{7} {s e c}^{7} x - \frac{1}{5} {s e c}^{5} x + c$ $⊛ M$
Commented by aliyn last updated on 07/Oct/21

$4) \int c o s 5 x . c o s 3 x d x = \frac{1}{2} \int (c o s 8 x + c o s 2 x) d x$ $= \frac{1}{16} s i n 8 x + \frac{1}{4} s i n 2 x + c$ $◻ M$
Commented by aliyn last updated on 07/Oct/21

$1) \int {s i n}^{5} x d x = \int ({s i n}^{2} x)^{2} s i n x d x$ $= \int {(1 - {c o s}^{2} x)}^{2} s i n x d x$ $= \int (1 - 2 {c o s}^{2} x + {c o s}^{4} x) s i n x d x$ $= \int (s i n x - 2 s i n x {c o s}^{2} x + s i n x {c o s}^{4} x) d x$ $= - c o s x + \frac{2}{3} {c o s}^{3} x - \frac{1}{5} {c o s}^{5} x + c$ $\oplus M$
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