What is the integral from x^5 to x^3 of (2t-1)^3 ?

The integral from x^5 to x^3 of (2t-1)^3 is x^3(-2x^{17}+4x^{12}+2x^9-3x^7-4x^6+3x^3+x^2-1)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from x^5 to x^3 of (2t-1)^3

\int_{x^{5}}^{x^{3}} (2 t - 1)^{3} d t

x^{3} (- 2 x^{17} + 4 x^{12} + 2 x^{9} - 3 x^{7} - 4 x^{6} + 3 x^{3} + x^{2} - 1)

Solution steps

\int_{x^{5}}^{x^{3}} (2 t - 1)^{3} d t

Apply u-substitution

= \int_{2 x^{5} - 1}^{2 x^{3} - 1} \frac{u ^{3}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{2 x^{5} - 1}^{2 x^{3} - 1} u^{3} d u

Apply the Power Rule

= \frac{1}{2} [\frac{u ^{4}}{4}]_{2 x^{5} - 1}^{2 x^{3} - 1}

Compute the boundaries:

\frac{( 2 x ^{3} - 1 ) ^{4}}{4} - \frac{( 2 x ^{5} - 1 ) ^{4}}{4}

= \frac{1}{2} (\frac{( 2 x ^{3} - 1 ) ^{4}}{4} - \frac{( 2 x ^{5} - 1 ) ^{4}}{4})

Simplify

= x^{3} (- 2 x^{17} + 4 x^{12} + 2 x^{9} - 3 x^{7} - 4 x^{6} + 3 x^{3} + x^{2} - 1)

What is the integral from x^5 to x^3 of (2t-1)^3 ?
The integral from x^5 to x^3 of (2t-1)^3 is x^3(-2x^{17}+4x^{12}+2x^9-3x^7-4x^6+3x^3+x^2-1)