\[ \int e^x \, dx = e^x + C \]

\[ \int e^{ax+b} \, dx = \frac{1}{a}e^{ax+b} + C \]

\[ \int a^x \, dx = \frac{a^x}{\ln a} + C \qquad (a>0,\,a\ne 1) \]

Let \(u = ax + b\). Then \(du = a\,dx\) → \(dx = \frac{du}{a}\). So:

\[ \int e^{ax+b} dx = \int e^u \cdot \frac{du}{a} = \frac{1}{a}e^u + C = \frac{1}{a}e^{ax+b} + C \]

Useful especially in definite integrals.

Example 1: \(\displaystyle \int e^{2x}\,dx\)

\[ \int e^{2x}dx = \frac{1}{2}e^{2x} + C \]

Example 2: \(\displaystyle \int 3e^{5x+1}\,dx\)

Factor the constant 3: \[ 3\int e^{5x+1}dx = 3\cdot\frac{1}{5}e^{5x+1} + C = \frac{3}{5}e^{5x+1} + C \]

Example 3: \(\displaystyle \int (2x+7)e^{x^2+7x}\,dx\)

Let \(u = x^2 + 7x\) → \(du = (2x + 7)dx\). \[ \int e^{x^2+7x}(2x+7)dx = \int e^u\,du = e^u + C = e^{x^2+7x} + C \]

Example 4 (Non-natural base): \(\displaystyle \int 4^x\,dx\)

\[ \int 4^x dx = \frac{4^x}{\ln 4} + C \]

Evaluate: \(\displaystyle \int_{0}^{2} e^{3x+1}\,dx\)

\[ \int e^{3x+1}dx = \frac{1}{3}e^{3x+1} \] So: \[ \left.\frac{1}{3}e^{3x+1}\right|_{0}^{2} = \frac{1}{3}\left(e^{7} - e^{1}\right) = \frac{e^7 - e}{3} \]

• If the exponent contains \(ax+b\), multiply by \(\frac{1}{a}\).
• If there is a factor of the derivative (like \(2x+7\)), substitution becomes instant.
• For \(a^x\), always divide by \(\ln a\).
• Combine exponential rules with partial fractions or substitution in hard questions.
• In definite integrals, change the limits if using substitution.