Question

Find the most general antiderivative of the function and check your answer by differentiation. f(x)=3*(e^x)+7*(sec^2)(x)

Answer 1

Indefinite integrals:

$$\int \left(3 e ^ {x} + 7 \sec^ {2} (x)\right) d x = 3 e ^ {x} + 7 \tan (x) + \text{constant}$$

Possible intermediate steps:

$$\int \left(3 e ^ {x} + 7 \sec^ {2} (x)\right) d x$$

Integrate the sum term by term and factor out constants:

$$= 3 \int e ^ {x} d x + 7 \int \sec^ {2} (x) d x$$

The integral of $\sec^2 (x)$ is $\tan (x)$:

$$= 7 \tan (x) + 3 \int e ^ {x} d x$$

The integral of $e^x$ is $e^x$:

$$= 3 e ^ {x} + 7 \tan (x) + \text{constant}$$

Check:

Derivative:

$$\frac {d}{d x} \left(3 e ^ {x} + 7 \tan (x)\right) = 3 e ^ {x} + 7 \sec^ {2} (x)$$

Possible derivation:

$$\frac {d}{d x} \left(3 e ^ {x} + 7 \tan (x)\right)$$

Differentiate the sum term by term and factor out constants:

$$= 3 \left(\frac {d}{d x} \left(e ^ {x}\right)\right) + 7 \left(\frac {d}{d x} (\tan (x))\right)$$

The derivative of $e^x$ is $e^x$:

$$= 7 \left(\frac {d}{d x} (\tan (x))\right) + 3 e ^ {x}$$

The derivative of $\tan (x)$ is $\sec^2 (x)$:

$$= 3 e ^ {x} + 7 \sec^ {2} (x)$$