Find the indefinite integral of ∫(5−5x+x5)dx

Question

Feas1981 · Accepted Answer

Step 1
We have to find the indefinite integral of the function:

\int (5 - 5^{x} + x^{5}) d x

We know the formula of integration,

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

\int d x = x + C

\int a^{x} d x = \frac{a^{x}}{\ln (a)} + C

Where, C is an arbitrary constant.
Step 2
Applying above formula for the given integral, we get

\int (5 - 5^{x} + x^{5}) d x = 5 \int d x - \int 5^{x} d x + \int x^{5} d x

= 5 x - \frac{5^{x}}{\ln (5)} + \frac{x^{5 + 1}}{5 + 1} + C

= 5 x - \frac{5^{x}}{\ln (5)} + \frac{x^{6}}{6} + C

Hence, value of the indefinite integral is

5 x - \frac{5^{x}}{\ln (5)} + \frac{x^{6}}{6} + C

.

Opeance1951 · Accepted Answer

Step 1: Expand.  ∫5−5x+x5dx  Step 2: Use Sum Rule: ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx.  ∫5+x5dx−∫5xdx  Step 3: Use Power Rule: ∫xndx=xn+1n+1+C.  5x+x66−∫5xdx  Step 4: Use this property: ∫axdx=axln⁡a.  5x+x66−5xln⁡5  Step 5: Add constant.  5x+x66−5xln⁡5+C

Find the indefinite integral of \int (5-5^{x}+x^{5})dx

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