Evaluate the integral by making the given substitution.∫x3(3+x4)7&nbsp;dx&nbsp;,u=3+x4

Step 1 ∫x3(3+x4)7dx,u=3+x4 substitute (3+x4)=u 4x3dx=∂u =∫x3(u)7∂u4x3 ==∫(u)74∂u Step 2 =14∫47∂u ∴∫af(x)dx=af(x)dx =14∫47+17+1 ∴∫x9dx=xa+1a+1a≠−1 ⇒ substitute (u=3+x4 =14(3+x4)88+c c is integration constant =132(3+x4)8+c

Expert answers to "Evaluate the integral by making the given substitution.∫x3(3+x4)7dx,u=3+x4"

Teddy Dillard

Answered question

2021-12-10

Evaluate the integral by making the given substitution. $\int x^{3} {(3 + x^{4})}^{7} d x, u = 3 + x^{4}$

Answer & Explanation

Bubich13

Beginner2021-12-11Added 36 answers

Step 1
$\int x^{3} {(3 + x^{4})}^{7} d x, u = 3 + x^{4}$
substitute $(3 + x^{4}) = u$
$4 x^{3} d x = \partial u$
$= \int x^{3} {(u)}^{7} \frac{\partial u}{4 x^{3}}$
$== \int \frac{{(u)}^{7}}{4} \partial u$
Step 2
$= \frac{1}{4} \int 4^{7} \partial u$ $∴ \int a f (x) d x = a f (x) d x$
$= \frac{1}{4} \int \frac{4^{7 + 1}}{7 + 1}$ $∴ \int x^{9} d x = \frac{x^{a + 1}}{a + 1} a \neq - 1$
$\Rightarrow$ substitute $(u = 3 + x^{4}$
$= \frac{1}{4} \frac{(3 + x^{4})^{8}}{8} + c$ c is integration constant
$= \frac{1}{32} {(3 + x^{4})}^{8} + c$

Rita Miller

Beginner2021-12-12Added 28 answers

I see no reason to write the solution a second time, it is decided unambiguously

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