Indefinite integral of e^(5x) via u-substitution

Evaluate the indefinite integral $\int e^{5x}\,dx$ by using the substitution $u = 5x$ .

Hint

Let $u = 5x$ , so $du = 5\,dx$ , which means $dx = \frac{1}{5}du$ . Substitute these into the integral to rewrite it entirely in terms of $u$ , then integrate and substitute back for $x$ .

Step-by-step solution

Let $u = 5x$ . Then $du = 5\,dx$ , so $dx = \frac{1}{5}\,du$ . Substituting into the integral gives $\int e^{5x}\,dx = \int e^{u}\cdot \frac{1}{5}\,du = \frac{1}{5}\int e^{u}\,du.$

The integral of $e^u$ is simply $e^u$ , so $\frac{1}{5}\int e^{u}\,du = \frac{1}{5}e^{u} + C$ . Substituting back $u = 5x$ gives the final answer $\int e^{5x}\,dx = \frac{1}{5}e^{5x} + C.$

Indefinite integral of e^(5x) via u-substitution

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