Evaluate the integrals. ∫x(x+3)10dx

Question

Evaluate the integrals.  ∫x(x+3)10dx

Parminquale · Accepted Answer

Step 1
According to the question, we have to integrate the given integral

\int x {(x + 3)}^{10} d x

.
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} d x = \frac{x^{n + 1}}{n + 1} + C

Step 2
Rewrite the given integral,

\int x {(x + 3)}^{10} d x

Now,
Let,

I = \int x {(x + 3)}^{10} d x

...(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 (t - 3) \cdot t^{10} d t

= \int t^{11} d t - \int 3 t^{10} d t

= \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{{(x + 3)}^{12}}{12} - \frac{3 {(x + 3)}^{11}}{11} + C

Sevensis1977 · Accepted Answer

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 {(x + 3)}^{10} d x

.

\int (u - 3) u^{10} d u

Step 3: Expand.

\int u^{11} - 3 u^{10} d u

Step 4: Use Power Rule:

\int x^{n} d x = \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{{(x + 3)}^{12}}{12} - \frac{3 {(x + 3)}^{11}}{11}

Step 6: Add constant.

\frac{{(x + 3)}^{12}}{12} - \frac{3 {(x + 3)}^{11}}{11} + C

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

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