Pearl Carney

2021-11-19

Evaluate the integrals.
$\int x{\left(x+3\right)}^{10}dx$

Parminquale

Step 1
According to the question, we have to integrate the given integral $\int x{\left(x+3\right)}^{10}dx$.
The given integral is indefinite integral and this type of integration have no fixed value that is why we add a constant value after the integration.
To solve the above integral, we have to use the power formula, which is given as follows,
$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$
Step 2
Rewrite the given integral,
$\int x{\left(x+3\right)}^{10}dx$
Now,
Let, $I=\int x{\left(x+3\right)}^{10}dx$...(1)
To solve further, we have to use substitution method, so substitutins x+3=t and proceeding as follows,
x+3=t...(1)
Differentiating both sides with respect to x, we get,
dx=dt
Now, substituting as x=(t−3) and dx=dt, in the equation (1), we get,
Step 3
$I=\int \left(t-3\right)\cdot {t}^{10}dt$
$=\int {t}^{11}dt-\int 3{t}^{10}dt$
$=\frac{{t}^{12}}{12}-\frac{3{t}^{11}}{11}+C$
Now, substitute back in the above answer as x+3=t,we get the final answer as,
$I=\frac{{\left(x+3\right)}^{12}}{12}-\frac{3{\left(x+3\right)}^{11}}{11}+C$

Sevensis1977

Step 1: Use Integration by Substitution.
Let u=x+3, du=dx then x dx=u-3du
Step 2: Using u and dudu above, rewrite $\int x{\left(x+3\right)}^{10}dx$.
$\int \left(u-3\right){u}^{10}du$
Step 3: Expand.
$\int {u}^{11}-3{u}^{10}du$
Step 4: Use Power Rule: $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$.
$\frac{{u}^{12}}{12}-\frac{3{u}^{11}}{11}$
Step 5: Substitute u=x+3 back into the original integral.
$\frac{{\left(x+3\right)}^{12}}{12}-\frac{3{\left(x+3\right)}^{11}}{11}$
$\frac{{\left(x+3\right)}^{12}}{12}-\frac{3{\left(x+3\right)}^{11}}{11}+C$