What is the integral from cos(x) to 5x of cos(u^3) ?

The integral from cos(x) to 5x of cos(u^3) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from cos(x) to 5x of cos(u^3)

\int_{c o s (x)}^{5 x} cos (u^{3}) d u

- \frac{5 x ( 3 i ( 5 x ) ^{3} Γ ( \frac{1}{3} , - i ( 5 x ) ^{3} ) + 3 - i ( 5 x ) ^{3} Γ ( \frac{1}{3} , i ( 5 x ) ^{3} ) )}{6 3 ( 5 x ) ^{6}} - - \frac{cos ( x ) ( 3 i cos ^{3} ( x ) Γ ( \frac{1}{3} , - i cos ^{3} ( x ) ) + 3 - i cos ^{3} ( x ) Γ ( \frac{1}{3} , i cos ^{3} ( x ) ) )}{6 3 cos ^{6} ( x )}

Solution steps

\int_{c o s (x)}^{5 x} cos (u^{3}) d u

Use the nonelementary integral:

\int cos (x^{3}) d x = - \frac{x ( 3 i x ^{3} Γ ( \frac{1}{3} , - i x ^{3} ) + 3 - i x ^{3} Γ ( \frac{1}{3} , i x ^{3} ) )}{6 3 x ^{6}}

= [- \frac{u ( 3 i u ^{3} Γ ( \frac{1}{3} , - i u ^{3} ) + 3 - i u ^{3} Γ ( \frac{1}{3} , i u ^{3} ) )}{6 3 u ^{6}}]_{c o s (x)}^{5 x}

Compute the boundaries:

- \frac{5 x ( 3 i ( 5 x ) ^{3} Γ ( \frac{1}{3} , - i ( 5 x ) ^{3} ) + 3 - i ( 5 x ) ^{3} Γ ( \frac{1}{3} , i ( 5 x ) ^{3} ) )}{6 3 ( 5 x ) ^{6}} - - \frac{cos ( x ) ( 3 i cos ^{3} ( x ) Γ ( \frac{1}{3} , - i cos ^{3} ( x ) ) + 3 - i cos ^{3} ( x ) Γ ( \frac{1}{3} , i cos ^{3} ( x ) ) )}{6 3 cos ^{6} ( x )}

= - \frac{5 x ( 3 i ( 5 x ) ^{3} Γ ( \frac{1}{3} , - i ( 5 x ) ^{3} ) + 3 - i ( 5 x ) ^{3} Γ ( \frac{1}{3} , i ( 5 x ) ^{3} ) )}{6 3 ( 5 x ) ^{6}} - - \frac{cos ( x ) ( 3 i cos ^{3} ( x ) Γ ( \frac{1}{3} , - i cos ^{3} ( x ) ) + 3 - i cos ^{3} ( x ) Γ ( \frac{1}{3} , i cos ^{3} ( x ) ) )}{6 3 cos ^{6} ( x )}

\int_{- 3}^{- 2} 2 x (5 - x^{2})^{3} d x

\int_{9}^{18} x x - 9 d x

\int_{\frac{π}{6}}^{\frac{π}{2}} (cos (x))^{2} d x

\int_{0}^{π} (5 e^{x} + 6 sin (x)) d x

\int_{0}^{1} x^{2} (x^{3} - 3)^{4} d x

What is the integral from cos(x) to 5x of cos(u^3) ?
The integral from cos(x) to 5x of cos(u^3) is