What is the integral of (6x^8)/(sec(10x^3)) ?

The integral of (6x^8)/(sec(10x^3)) is 1/250 (50x^6sin(10x^3)-sin(10x^3)+10x^3cos(10x^3))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (6x^8)/(sec(10x^3))

\int \frac{6 x ^{8}}{sec ( 10 x ^{3} )} d x

\frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3})) + C

Solution steps

\int \frac{6 x ^{8}}{sec ( 10 x ^{3} )} d x

Take the constant out:

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

= 6 \cdot \int \frac{x ^{8}}{sec ( 10 x ^{3} )} d x

Apply u-substitution

: \int \frac{u ^{2}}{3000 sec ( u )} d u

= 6 \cdot \int \frac{u ^{2}}{3000 sec ( u )} d u

Take the constant out:

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

= 6 \cdot \frac{1}{3000} \cdot \int \frac{u ^{2}}{sec ( u )} d u

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

= 6 \cdot \frac{1}{3000} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

: u^{2} sin (u) - \int 2 u sin (u) d u

= 6 \cdot \frac{1}{3000} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= 6 \cdot \frac{1}{3000} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = 10 x^{3}

= 6 \cdot \frac{1}{3000} ((10 x^{3})^{2} sin (10 x^{3}) - 2 (- 10 x^{3} cos (10 x^{3}) + sin (10 x^{3})))

Simplify

6 \cdot \frac{1}{3000} ((10 x^{3})^{2} sin (10 x^{3}) - 2 (- 10 x^{3} cos (10 x^{3}) + sin (10 x^{3}))) : \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3}))

= \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3}))

Add a constant to the solution

= \frac{1}{250} (50 x^{6} sin (10 x^{3}) - sin (10 x^{3}) + 10 x^{3} cos (10 x^{3})) + C

\int 3^{2 x - 1} d x

\int (cos^{3} (3 x) - cos (3 x)) d x

\int (x + 2) x^{2} + 4 x + 1 d x

\int (3 e^{x} + \frac{4}{x}) d x

\int 9 x^{2} + 4 d x

What is the integral of (6x^8)/(sec(10x^3)) ?
The integral of (6x^8)/(sec(10x^3)) is 1/250 (50x^6sin(10x^3)-sin(10x^3)+10x^3cos(10x^3))+C