Evaluate the integral: integral (2^x + 5^x)² dx
Evaluate the integral: $$ \int \left(2^x + 5^x\right)^2 \, dx. $$
Explanation of Method
Expanding $(a^x+b^x)^2$ using $(a+b)^2 = a^2+2ab+b^2$ produces three exponential terms. Each squared term, like $(2^x)^2$, is itself a single exponential with base $a^2$, and the cross term $2\cdot a^x b^x$ combines into $2(ab)^x$. Once every term is a pure exponential, the general antiderivative formula applies to each.
- Expand the square using the binomial identity.
- Simplify each term into a single exponential base.
- Integrate each exponential term separately.
- Combine the results and add the constant of integration.
Step 1 · Expand the square
Using the algebraic identity $(a+b)^2 = a^2+2ab+b^2$, the expression $\left(2^x+5^x\right)^2$ expands to:
$$ \left(2^x+5^x\right)^2 = (2^x)^2 + 2(2^x)(5^x) + (5^x)^2. $$
Simplify each term:
$$ (2^x)^2 = 2^{2x}, \qquad 2(2^x)(5^x) = 2\cdot 2^x\cdot 5^x = 2\cdot(2\cdot 5)^x = 2\cdot 10^x, \qquad (5^x)^2 = 5^{2x}. $$
Thus, the integrand becomes:
$$ \left(2^x+5^x\right)^2 = 2^{2x} + 2\cdot 10^x + 5^{2x}. $$
Step 2 · Rewrite the integral
The integral can now be written as:
$$ \int \left(2^x+5^x\right)^2 dx = \int 2^{2x}\,dx + \int 2\cdot 10^x\,dx + \int 5^{2x}\,dx. $$
Step 3 · Solve each term
We evaluate each integral separately using the formula for the integral of an exponential function:
$$ \int a^x\,dx = \frac{a^x}{\ln(a)} + C, \quad \text{for } a>0,\ a\neq 1. $$
First term: $\displaystyle \int 2^{2x}\,dx$. Rewrite $2^{2x}$ as $(2^2)^x = 4^x$. Then:
$$ \int 2^{2x}\,dx = \int 4^x\,dx = \frac{4^x}{\ln(4)}. $$
Second term: $\displaystyle \int 2\cdot 10^x\,dx$. Using the constant multiple rule, $\int 2\cdot 10^x\,dx = 2\int 10^x\,dx$. Solve the integral:
$$ \int 10^x\,dx = \frac{10^x}{\ln(10)}. $$
So:
$$ \int 2\cdot 10^x\,dx = 2\cdot \frac{10^x}{\ln(10)} = \frac{2\cdot 10^x}{\ln(10)}. $$
Third term: $\displaystyle \int 5^{2x}\,dx$. Rewrite $5^{2x}$ as $(5^2)^x = 25^x$. Then:
$$ \int 5^{2x}\,dx = \int 25^x\,dx = \frac{25^x}{\ln(25)}. $$
Step 4 · Combine the results
Adding the results of the three terms, we have:
$$ \int \left(2^x+5^x\right)^2 dx = \frac{4^x}{\ln(4)} + \frac{2\cdot 10^x}{\ln(10)} + \frac{25^x}{\ln(25)} + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Rewritten as | Antiderivative |
|---|---|---|
| $(2^x)^2$ | $4^x$ | $\dfrac{4^x}{\ln 4}$ |
| $2\cdot 2^x\cdot 5^x$ | $2\cdot 10^x$ | $\dfrac{2\cdot 10^x}{\ln 10}$ |
| $(5^x)^2$ | $25^x$ | $\dfrac{25^x}{\ln 25}$ |
Additional Notes
- The integrand is defined for all real $x$, since exponential functions with positive base are defined everywhere.
- The cross term $2\cdot 2^x\cdot 5^x$ combines because $2^x\cdot 5^x = (2\cdot 5)^x = 10^x$, a direct consequence of the exponent rule $a^x b^x = (ab)^x$.
- The result can be checked by differentiating $\dfrac{4^x}{\ln 4}+\dfrac{2\cdot 10^x}{\ln 10}+\dfrac{25^x}{\ln 25}$ and confirming it returns the original integrand.
Result
The solution to the integral is: