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受欢迎的微积分 >

integral of sec^4(xta)n^5x

解答

\int sec^{4} (x t a) n^{5} x d x

解答

n^{5} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + \frac{1}{3} a t x tan^{3} (a t x) + ln ∣ cos (a t x) ∣ + \frac{1}{3} ln ∣ sec (a t x) ∣ - \frac{1}{6} sec^{2} (a t x)) + C

求解步骤

\int sec^{4} (x t a) n^{5} x d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= n^{5} \cdot \int sec^{4} (a t x) x d x

使用换元积分法

= n^{5} \cdot \int \frac{u sec ^{4} ( u )}{a ^{2} t ^{2}} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= n^{5} \frac{1}{a ^{2} t ^{2}} \cdot \int u sec^{4} (u) d u

使用分布积分法

= n^{5} \frac{1}{a ^{2} t ^{2}} (u (tan (u) + \frac{1}{3} tan^{3} (u)) - \int tan (u) + \frac{1}{3} tan^{3} (u) d u)

\int tan (u) + \frac{1}{3} tan^{3} (u) d u = - ln ∣ cos (u) ∣ + \frac{1}{3} (- ln ∣ sec (u) ∣ + \frac{sec ^{2} ( u )}{2})

= n^{5} \frac{1}{a ^{2} t ^{2}} (u (tan (u) + \frac{1}{3} tan^{3} (u)) - (- ln ∣ cos (u) ∣ + \frac{1}{3} (- ln ∣ sec (u) ∣ + \frac{sec ^{2} ( u )}{2})))

u = a t x

代回

= n^{5} \frac{1}{a ^{2} t ^{2}} (a t x (tan (a t x) + \frac{1}{3} tan^{3} (a t x)) - (- ln ∣ cos (a t x) ∣ + \frac{1}{3} (- ln ∣ sec (a t x) ∣ + \frac{sec ^{2} ( a t x )}{2})))

化简

n^{5} \frac{1}{a ^{2} t ^{2}} (a t x (tan (a t x) + \frac{1}{3} tan^{3} (a t x)) - (- ln ∣ cos (a t x) ∣ + \frac{1}{3} (- ln ∣ sec (a t x) ∣ + \frac{sec ^{2} ( a t x )}{2}))) : n^{5} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + \frac{1}{3} a t x tan^{3} (a t x) + ln ∣ cos (a t x) ∣ + \frac{1}{3} ln ∣ sec (a t x) ∣ - \frac{1}{6} sec^{2} (a t x))

= n^{5} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + \frac{1}{3} a t x tan^{3} (a t x) + ln ∣ cos (a t x) ∣ + \frac{1}{3} ln ∣ sec (a t x) ∣ - \frac{1}{6} sec^{2} (a t x))

解答补常数

= n^{5} \frac{1}{a ^{2} t ^{2}} (a t x tan (a t x) + \frac{1}{3} a t x tan^{3} (a t x) + ln ∣ cos (a t x) ∣ + \frac{1}{3} ln ∣ sec (a t x) ∣ - \frac{1}{6} sec^{2} (a t x)) + C

作图

流行的例子

sum from n=1 to infinity of 4/(n(n+7))

n = 1 \sum \infty \frac{4}{n ( n + 7 )}

sum from n=2 to infinity of n/(n+2)

n = 2 \sum \infty \frac{n}{n + 2}

derivative f(x)=(x^2+3x-2)/(x^2-5x+6)

d er i v a t i v e f (x) = \frac{x ^{2} + 3 x - 2}{x ^{2} - 5 x + 6}

derivative f(x)=e^x-x^3

d er i v a t i v e f (x) = e^{x} - x^{3}

y^{''}+12y^'+36y=0,y(1)=0,y^'(1)=-1

y^{^{''}} + 12 y^{^{'}} + 36 y = 0, y (1) = 0, y^{^{'}} (1) = - 1